Matrix approach to noncommutative stably free modules and Hermite rings
In this paper we present a matrix-constructive proof of an Stafford’s Theorem about stably free modules over noncommutative rings. Matrix characterizations of noncommutative Hermite and projective-free rings are exhibit. Quotients, products and localizations of Hermite and some other classes of ring...
Збережено в:
| Дата: | 2014 |
|---|---|
| Автори: | , |
| Формат: | Стаття |
| Мова: | English |
| Опубліковано: |
Інститут прикладної математики і механіки НАН України
2014
|
| Назва видання: | Algebra and Discrete Mathematics |
| Онлайн доступ: | http://dspace.nbuv.gov.ua/handle/123456789/153350 |
| Теги: |
Додати тег
Немає тегів, Будьте першим, хто поставить тег для цього запису!
|
| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | Matrix approach to noncommutative stably free modules and Hermite rings / O. Lezama, C. Gallego // Algebra and Discrete Mathematics. — 2014. — Vol. 18, № 1. — С. 109–137. — Бібліогр.: 26 назв. — англ. |
Репозитарії
Digital Library of Periodicals of National Academy of Sciences of Ukraine| id |
irk-123456789-153350 |
|---|---|
| record_format |
dspace |
| spelling |
irk-123456789-1533502019-06-15T01:30:38Z Matrix approach to noncommutative stably free modules and Hermite rings Lezama, O. Gallego, C. In this paper we present a matrix-constructive proof of an Stafford’s Theorem about stably free modules over noncommutative rings. Matrix characterizations of noncommutative Hermite and projective-free rings are exhibit. Quotients, products and localizations of Hermite and some other classes of rings close related to Hermite rings are also considered. 2014 Article Matrix approach to noncommutative stably free modules and Hermite rings / O. Lezama, C. Gallego // Algebra and Discrete Mathematics. — 2014. — Vol. 18, № 1. — С. 109–137. — Бібліогр.: 26 назв. — англ. 1726-3255 2010 MSC:16D40; 15A21. http://dspace.nbuv.gov.ua/handle/123456789/153350 en Algebra and Discrete Mathematics Інститут прикладної математики і механіки НАН України |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| language |
English |
| description |
In this paper we present a matrix-constructive proof of an Stafford’s Theorem about stably free modules over noncommutative rings. Matrix characterizations of noncommutative Hermite and projective-free rings are exhibit. Quotients, products and localizations of Hermite and some other classes of rings close related to Hermite rings are also considered. |
| format |
Article |
| author |
Lezama, O. Gallego, C. |
| spellingShingle |
Lezama, O. Gallego, C. Matrix approach to noncommutative stably free modules and Hermite rings Algebra and Discrete Mathematics |
| author_facet |
Lezama, O. Gallego, C. |
| author_sort |
Lezama, O. |
| title |
Matrix approach to noncommutative stably free modules and Hermite rings |
| title_short |
Matrix approach to noncommutative stably free modules and Hermite rings |
| title_full |
Matrix approach to noncommutative stably free modules and Hermite rings |
| title_fullStr |
Matrix approach to noncommutative stably free modules and Hermite rings |
| title_full_unstemmed |
Matrix approach to noncommutative stably free modules and Hermite rings |
| title_sort |
matrix approach to noncommutative stably free modules and hermite rings |
| publisher |
Інститут прикладної математики і механіки НАН України |
| publishDate |
2014 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/153350 |
| citation_txt |
Matrix approach to noncommutative stably free modules and Hermite rings / O. Lezama, C. Gallego // Algebra and Discrete Mathematics. — 2014. — Vol. 18, № 1. — С. 109–137. — Бібліогр.: 26 назв. — англ. |
| series |
Algebra and Discrete Mathematics |
| work_keys_str_mv |
AT lezamao matrixapproachtononcommutativestablyfreemodulesandhermiterings AT gallegoc matrixapproachtononcommutativestablyfreemodulesandhermiterings |
| first_indexed |
2025-07-14T04:34:25Z |
| last_indexed |
2025-07-14T04:34:25Z |
| _version_ |
1837595540235223040 |
| fulltext |
Algebra and Discrete Mathematics RESEARCH ARTICLE
Volume 18 (2014). Number 1, pp. 109 – 137
© Journal “Algebra and Discrete Mathematics”
Matrix approach to noncommutative
stably free modules and Hermite rings
Oswaldo Lezama and Claudia Gallego
Communicated by V. A. Artamonov
Abstract. In this paper we present a matrix-constructive
proof of an Stafford’s Theorem about stably free modules over
noncommutative rings. Matrix characterizations of noncommutative
Hermite and projective-free rings are exhibit. Quotients, products
and localizations of Hermite and some other classes of rings close
related to Hermite rings are also considered.
1. Introduction
Finitely generated projective modules, stably free modules, projective-
free rings, Bézout rings and Hermite rings have recently encountered
interesting applications in algebraic control theory and algebraic analysis.
For example, in [19], internal stabilization of coherent control systems
over a commutative domain S is characterized in terms of conditions on
S as being a Prüfer domain or an Hermite commutative ring (see [19],
Theorems 2 and 4). In [20], the concept of internal stabilizability of stable
time-invariant linear system over S is equivalent to the fact that certain
S-modules are projective, whereas the existence of a doubly coprime
factorization corresponds to the freeness of the same modules. In noncom-
mutative algebraic analysis these homological objects have been applied
to describe functional linear system of differential equations (SLF) over
Ore algebras using matrix interpretations ([5], [6], [7], [18]).
2010 MSC: 16D40; 15A21.
Key words and phrases: noncommutative rings and modules, stably free modu-
les, Hermite rings, matrix methods in homological algebra.
110 Matrix approach to Hermite rings
From a computational approach, it is very useful to study projective
modules, stably free modules, projective-free rings and Hermite rings
from a matrix-constructive point of view. This was done in [15] for the
commutative case, the present paper can be considered as a generalization
from commutative to noncommutative rings of some results presented
in [15], Chapter 6 (see also [14]). The first section, needed for the rest of
the paper, is about elementary facts of linear algebra over noncommutative
rings and stably free modules (see also [8] and [17]). The second section
is dedicated to give a matrix-constructive proof of a theorem due Stafford
about stably free modules. The proof has been adapted from [18] but we
can avoid the involution used in [18]. Matrix characterizations of Hermite
and projective-free rings are presented in Sections 3 and 4. A matrix proof
of a Kaplansky theorem about finitely generated projective modules over
local rings is also included. Some remarkable classes of rings closed related
to Hermite rings are considered in Section 5, in particular, we prove that
for rings without nontrivial idempotents projective-free rings coincide
with ID rings (i.e., rings for which each idempotent matrix is similar to a
Smith normal diagonal matrix). Moreover, we will see that every ID ring
S is Hermite when all idempotents of S are central. Products, quotients
and localizations of Hermite rings are studied in Section 6, we proved
that if S is a left Noetherian ring and P is a prime (completely prime)
left localizable ideal of S, then SP is Hermite (projective-free).
1.1. Some topics from linear algebra
We start recalling some notations and well known elementary prop-
erties of linear algebra for left modules. All rings are noncommutative
and modules will be considered on the left; S will represent an arbitrary
noncommutative ring; Sr is the left S-module of columns of size r × 1;
if Ss f−→ Sr is an S-homomorphism then there is a matrix associated to f
in the canonical bases of Sr and Ss, denoted F := m(f), and disposed by
columns, i.e., F ∈ Mr×s(S). In fact, if f is given by Ss f−→ Sr , ej 7→ f j ,
where {e1, . . . , es} is the canonical basis of Ss, f can be represented
by a matrix, i.e., if f j :=
[
f1j . . . frj
]T
, then the matrix of f in the
canonical bases of Ss and Sr is
F :=
[
f 1 · · · f s
]
=
f11 · · · f1s
...
...
fr1 · · · frs
∈ Mr×s(S).
O. Lezama, C. Gallego 111
Observe that Im(f) is the column module of F , i.e., the left S-module
generated by the columns of F , denoted by 〈F 〉:
Im(f) = 〈f(e1), . . . , f(es)〉 = 〈f 1, . . . , f s〉 = 〈F 〉.
We recall also that
Syz({f 1, . . . , f s}) := {a :=
[
a1 · · · as
]T
∈ Ss|a1f 1 + · · · + asf s = 0},
and
Syz({f 1, . . . , f s}) = ker(f). (1)
Moreover, observe that if a := (a1, . . . , as)T ∈ Ss, then
f(a) = (aT F T )T . (2)
Note that function m : HomS(Ss, Sr) → Mr×s(S) is bijective; moreover,
if Sr g−→ Sp is a homomorphism, then the matrix of gf in the canonical
bases is m(gf) = (F T GT )T . Thus, f : Sr → Sr is an isomorphism if
and only if F T ∈ GLr(S). Finally, let C ∈ Mr(S); the columns of C
conform a basis of Sr if and only if CT ∈ GLr(S). With this notation, a
matrix characterization of f.g. projective modules can be formulated in
the following way.
Proposition 1. Let S be an arbitrary ring and M a S-module. Then,
M is a f.g. projective S-module if and only if there exists a square matrix
F over S such that F T is idempotent and M = 〈F 〉.
Proof. ⇒): If M = 0, then F = 0; let M 6= 0, there exists s > 1 and a
M ′ such that Ss = M ⊕ M ′; let f : Ss → Ss be the projection on M
and F the matrix of f in the canonical basis of Ss. Then, f2 = f and
(F T F T )T = F , so F T F T = F T ; note that M = Im(f) = 〈F 〉.
⇐): Let f : Ss → Ss be the homomorphism defined by F (see (2));
from F T F T = F T we get that f2 = f , moreover, since M = 〈F 〉, then
Im(f) = M and hence M is direct summand of Ss, i.e., M is f.g. projective
(observe that the complement M ′ of M is ker(f) and f is the projection
on M).
Remark 1. (i) When S is commutative, or when we consider right mo-
dules instead of left modules, (2) says that f(a) = Fa. Moreover, the
matrix of a compose homomorphism gf is given by m(gf) = m(g)m(f).
Note that f : Sr → Sr is an isomorphism if and only if F ∈ GLr(S);
moreover, C ∈ GLr(S) if and only if its columns conform a basis of Sr. In
addition, Proposition 1 says that M is a f.g. projective S-module if and
112 Matrix approach to Hermite rings
only if there exists a square matrix F over S such that F is idempotent
and M = 〈F 〉.
(ii) When the matrices of homomorphisms of left modules are disposed
by rows instead of by columns, i.e., if S1×s is the left free module of rows
vectors of length s and the matrix of the homomorphism S1×s f−→ S1×r is
defined by
F ′ =
f ′
11 · · · f ′
1r
...
...
f ′
s1 · · · f ′
sr
:=
f11 · · · fr1
...
...
f1s · · · frs
∈ Ms×r(S),
then
f(a1, . . . , as) = (a1, . . . , as)F ′, (3)
i.e., f(aT ) = aT F T . Thus, the values given by (3) and (2) agree since
F ′ = F T . Moreover, the composed homomorphism gf means that g acts
first and then acts f , and hence, the matrix of gf is given by m(gf) =
m(g)m(f). Note that f : S1×r → S1×r is an isomorphism if and only if
m(f) ∈ GLr(S); moreover, C ∈ GLr(S) if and only if its rows conform a
basis of S1×r. This left-row notation is also used in [8] and [18]. Observe
that with this notation, the proof of Proposition 1 says that M is a f.g.
projective S-module if and only if there exists a square matrix F over S
such that F is idempotent and M = 〈F 〉, but in this case 〈F 〉 represents
the module generated by the rows of F . Note that Proposition 1 could
has been formulated this way: In fact, the set of idempotents matrices of
Ms(S) coincides with the set {F T |F ∈ Ms(S), F T idempotent}.
Definition 1 ([12]). Let S be a ring.
(i) S satisfies the rank condition (RC) if for any integers r, s > 1, given
an epimorphism Sr f−→ Ss, then r > s.
(ii) S is an IBN ring (Invariant Basis Number) if for any integers
r, s > 1, Sr ∼= Ss if and only if r = s.
Proposition 2. Let S be a ring.
(i) S is RC if and only if given any matrix F ∈ Ms×r(S) the following
condition holds:
if F has a right inverse then r > s.
(ii) S is RC if and only if given any matrix F ∈ Ms×r(S) the following
condition holds:
if F has a left inverse then s > r.
O. Lezama, C. Gallego 113
Proof. (i) ⇒): Let G be a right inverse of F , FG = Is; let f : Sr → Ss and
g : Ss → Sr such that m(f) = F and m(g) = G. Then, ((F T )T (GT )T )T =
Is; let fT : Ss → Sr and gT : Sr → Ss such that m(fT ) = F T and
m(gT ) = GT , then m(gT fT ) = m(iSs) and hence gT fT = iSs , i.e., gT is
surjective. Since S is RC, then r > s.
⇐): Let Sr f−→ Ss be an epimorphism, there exists Ss g−→ Sr such that
fg = iSs ; let F := m(f) ∈ Ms×r(S) and G := m(g) ∈ Mr×s(S), then
m(fg) = (GT F T )T = Is, so GT F T = Is, i.e., GT has right inverse, and
by hypothesis r > s. This means that S is RC.
(ii) ⇒): Let G ∈ Mr×s(S) a left inverse of F , then G has right inverse,
and by (i), s > r.
⇐): Let Sr f−→ Ss be an epimorphism; as in (i), GT F T = Is, so
F T ∈ Mr×s(S) has a left inverse and by the hypothesis r > s. Thus, S
is RC.
Proposition 3. RC ⇒ IBN .
Proof. Let Sr f−→ Ss be an isomorphism, then f is an epimorphism, and
hence r > s; considering f−1 we get that s > r.
Remark 2. Most of rings are RC, and hence, IBN . For example, com-
mutative rings and left Noetherian rings are RC (see [17] and [9]). The
condition IBN for rings is independent of the side we are considering the
modules (see [8]). The same is true for the RC property. From now on we
will assume that all rings considered in the present paper are RC.
1.2. Stably free modules
Definition 2. Let M be an S-module and t > 0 an integer. M is stably
free of rank t > 0 if there exist an integer s > 0 such that Ss+t ∼= Ss ⊕ M .
The rank of M is denoted by rank(M). Note that any stably free
module M is finitely generated and projective. Moreover, as we will show
in the next proposition, rank(M) is well defined, i.e., rank(M) is unique
for M .
Proposition 4. Let t, t′, s, s′ > 0 integers such that Ss+t ∼= Ss ⊕ M and
Ss′+t′ ∼= Ss′ ⊕ M . Then, t′ = t.
Proof. We have Ss′ ⊕Ss+t ∼= Ss′ ⊕Ss ⊕M and Ss ⊕Ss′+t′ ∼= Ss ⊕Ss′ ⊕M ,
then since S is an IBN ring, s′ + s + t = s + s′ + t′, and hence t′ = t.
114 Matrix approach to Hermite rings
Corollary 1. M is stably free of rank t > 0 if and only if there exist
integers r, s > 0 such that Sr ∼= Ss ⊕ M , with r > s and t = r − s.
Proof. If M is stably free of rank t, then Ss+t ∼= Ss ⊕M for some integers
s, t > 0, then taking r := s + t we get the result. Conversely, if there exist
integers r, s > 0 such that Sr ∼= Ss⊕M , with r > s, then Ss+r−s ∼= Ss⊕M ,
i.e., M is stably free of rank r − s.
Proposition 5. Let M be an S-module and let r, s > 0 integers such
that Sr ∼= Ss ⊕ M . Then r > s.
Proof. The canonical projection Sr → Ss is an epimorphism, but since
we are assuming that S is RC, then r > s.
Corollary 2. M is stably free if and only if there exist integers r, s > 0
such that Sr ∼= Ss ⊕ M .
Proof. This is a direct consequence of Corollary 1 and Proposition 5.
Proposition 6. Let M be an S-module. Then, the following conditions
are equivalent
(i) M is stably free.
(ii) M has a minimal presentation, i.e., M has a finite presentation
Ss f1−→ Sr f0−→ M → 0, where f1 has a left inverse.
Proof. See [17], Chapter 11.
From this proposition we get a matrix characterization of stably free
modules (compare with [18], Lemma 16).
Corollary 3. Let M be an S-module. Then the following conditions are
equivalent:
(i) M is stably free.
(ii) M is projective and has a finite system of generators f1, . . . , fr such
that Syz{f1, . . . , fr} is the module generated by the columns of a
matrix F1 of size r × s such that F T
1 has a right inverse.
Proof. (i) ⇒ (ii) By Proposition 6, M is projective and has a finite
presentation Ss f1−→ Sr f0−→ M → 0, where f1 has a left inverse. Let
f i = f0(ei), where {ei}16i6r is the canonical basis of Sr. Then M =
〈f 1, . . . , f r〉 and Im(f1) = ker(f0) = Syz{f 1, . . . , f r}, but Im(f1) is the
O. Lezama, C. Gallego 115
module generated by the columns of the matrix F1 defined by f1 in the
canonical bases. Thus, let g1 : Sr → Ss be a left inverse of f1, then
g1f1 = iSs and the matrix of g1f1 in the canonical bases is Is = (F T
1 GT
1 )T ,
so Is = F T
1 GT
1 .
(ii) ⇒ (i) Let f 1, . . . , f r be a set of generators of M such that
Syz{f 1, . . . , f r} is the module generated by the columns of a matrix
F1 of size r × s such that F T
1 has a right inverse. We have the exact
sequence 0 → ker(f0)
ι−→ Sr f0−→ M → 0, where ι is the canonical injection
and f0 is defined as above. We have ker(f0) = Syz{f 1, . . . , f r} = 〈F1〉,
and thus we get the finite presentation Ss f1−→ Sr f0−→ M → 0, where f1(ej)
is the jth column of F1, 1 6 j 6 s. By hypothesis F T
1 has a right inverse,
F T
1 GT
1 = Is, so Is = (F T
1 GT
1 )T . Let g1 : Sr → Ss be the homomorphism
defined by G1 ∈ Ms×r(S) in the canonical bases, then g1f1 = iSs and f1
is injective, this implies that the sequence 0 → Ss f1−→ Sr f0−→ M → 0 is
exact. By Proposition 6, M is stably free.
2. Stafford’s theorem: a constructive proof
A well known result due Stafford says that any left ideal of the Weyl
algebras D := An(K) or Bn(K), where K is a filed with char(K) = 0,
is generated by two elements. Recall that the Weyl algebra is defined
by An(K) := K[t1, . . . , tn][x1; ∂/∂t1] · · · [xn; ∂/∂tn] and Bn(K) is the
extended Weyl algebra defined as K(t1, . . . , tn)[x1; ∂/∂t1] · · · [xn; ∂/∂tn]
(see [22] and [18]). From the Stafford’s Theorem follows that any stably
free left module M over D with rank(M) > 2 is free. In [18] is presented
a constructive proof of this result that we want to study for arbitrary RC
rings. Actually, we will consider the generalization given in [18] staying
that any stably free left S-module M with rank(M) > sr(S) is free, where
sr(S) denotes the stable rank of the ring S. Our proof have been adapted
from [18], however we do not need the involution of ring S used in [18]
because of our left notation for modules and column representation for
homomorphism. This could justify our special left-column notation.
Definition 3. Let F be a matrix over S of size r × s. Then,
(i) Let r > s. F is unimodular if and only if F has a left inverse.
(ii) Let s > r. F is unimodular if and only if F has a right inverse.
In particular, the set of unimodular column matrices of size r × 1 is
denoted by Umc(r, S). Umr(s, S) is the set of unimodular row matrices
of size 1 × s.
116 Matrix approach to Hermite rings
Note that a column matrix is unimodular if and only if the left ideal
generated by its entries coincides with S, and a row matrix is unimodular
if and only if the right ideal generated by its entries is S.
Definition 4. Let S be a ring and v :=
[
v1 . . . vr
]T
∈ Umc(r, S) an
unimodular column vector. v is called stable (reducible) if there exists
a1, . . . , ar−1 ∈ S such that v′ :=
[
v1 + a1vr . . . vr−1 + ar−1vr
]T
is
unimodular. It says that the left stable rank of S is d > 1, denoted
sr(S) = d, if d is the least positive integer such that every unimodular
column vector of length d + 1 is stable. It says that sr(S) = ∞ if for every
d > 1 there exits a non stable unimodular column vector of length d + 1.
In a similar way is defined the right stable rank of S, however, both
ranks coincide ([2]). Two preliminary results are needed for the main
theorem of this section.
Proposition 7. Let S be a ring and v :=
[
v1 . . . vr
]T
an unimodu-
lar stable column vector over S, then there exists U ∈ Er(S) such that
Uv = e1.
Proof. See [18], Proposition 38.
Lemma 1. Let S be a ring and M a stably free S-module given by a
minimal presentation Ss f1−→ Sr f0−→ M → 0. Let g1 : Sr → Ss such that
g1f1 = iSs. Then the following conditions are equivalent:
(i) M is free of dimension r − s.
(ii) There exists a matrix U ∈ GLr(S) such that UGT
1 =
[
Is
0
]
, where
G1 is the matrix of g1 in the canonical bases. In such case, the last
r − s columns of UT conform a basis for M . Moreover, the first s
columns of UT conform the matrix F1 of f1 in the canonical bases.
(iii) There exists a matrix V ∈ GLr(S) such that GT
1 coincides with
the first s columns of V , i.e., GT
1 can be completed to an invertible
matrix V of GLr(S).
Proof. By the hypothesis, the exact sequence 0 → Ss f1−→ Sr f0−→ M → 0
splits, so F T
1 admits a right inverse GT
1 , where F1 is the matrix of f1 in
the canonical bases and G1 is the matrix of g1 : Sr → Ss, with g1f1 = iSs ,
i.e., F T
1 GT
1 = Is. Moreover, there exists g0 : M → Sr such that f0g0 = iM .
O. Lezama, C. Gallego 117
From this we get also the split sequence 0 → M
g0−→ Sr g1−→ Ss → 0. Note
that M ∼= ker(g1).
(i) ⇒ (ii): We have Sr = ker(g1) ⊕ Im(f1); by the hypothesis ker(g1)
is free. If s = r then ker(g1) = 0 and hence f1 is an isomorphism, so
f1g1 = iSs , i.e., GT
1 F T
1 = Is. Thus, we can take U := F T
1 .
Let r > s; if {e1, . . . , es} is the canonical basis of Ss, then {u1, . . . , us} is
a basis of Im(f1) with ui := f1(ei), 1 6 i 6 s; let {v1, . . . , vp} be a basis
of ker(g1) with p = r − s. Then, {v1, . . . , vp, u1, . . . , us} is a basis of Sr.
We define Sr h−→ Sr by h(ei) := ui for 1 6 i 6 s, and h(es+j) := vj for
1 6 j 6 p. Clearly h is bijective; moreover, g1h(ei) = g1(ui) = g1f1(ei) =
ei and g1h(es+j) = g1(vj) = 0, i.e., HT GT
1 =
[
Is
0
]
. Let U := HT , so we
observe that the last p columns of UT conform a basis of ker(g1) ∼= M
and the first s columns of UT conform F1.
(ii) ⇒ (i): Let U(k) the k-th row of U , then we have that UGT
1 =
[U(1) · · · U(s) · · · U(r)]
T GT
1 =
[
Is
0
]
, so U(i)G
T
1 = eT
i , 1 6 i 6 s, U(s+j)G
T
1 =
0, 1 6 j 6 p with p := r − s. This means that (U(s+j))
T ∈ ker(g1)
and hence 〈(U(s+j))
T |1 6 j 6 p〉 ⊆ ker(g1). On the other hand, let c ∈
ker(g1) ⊆ Sr, then cT GT
1 = 0 and cT U−1UGT
1 = 0, thus cT U−1
[
Is
0
]
=
0 and hence (cT U−1)T ∈ ker(l), where l : Sr → Ss is the homo-
morphism with matrix
[
Is 0
]
. Let d = [d1, . . . , dr]T ∈ ker(l), then
[d1, . . . , dr]
[
Is
0
]
= 0 and from this we conclude that d1 = · · · = ds = 0,
i.e., ker(l) = 〈es+1, es+2, . . . , es+p〉. From (cT U−1)T ∈ ker(l) we get that
(cT U−1)T = a1·es+1+· · ·+ap·es+p, so cT U−1 = (a1·es+1+· · ·+ap·es+p)T ,
i.e., cT = (a1 · es+1 + · · · + ap · es+p)T U and from this we get that c ∈
〈(U(s+j))
T |1 6 j 6 p〉. This proves that ker(g1) = 〈(U(s+j))
T |1 6 j 6 p〉;
but since U is invertible, then ker(g1) is free of dimension p. We have
proved also that the last p columns of UT conform a basis for ker(g1) ∼= M .
(ii) ⇔ (iii): UGT
1 =
[
Is
0
]
if and only if GT
1 = U−1
[
Is
0
]
, but the first
s columns of U−1
[
Is
0
]
coincides with the first s columns of U−1; taking
V := U−1 we get the result.
118 Matrix approach to Hermite rings
Theorem 1. Let S be a ring. Then any stably free S-module M with
rank(M) > sr(S) is free with dimension equals to rank(M).
Proof. Since M is stably free it has a minimal presentation, and hence,
it is given by an exact sequence
0 → Ss f1−→ Sr f0−→ M → 0;
moreover, note that rank(M) = r − s. Since this sequence splits, F T
1
admits a right inverse GT
1 , where F1 is the matrix of f1 in the canonical
bases and G1 is the matrix of g1 : Sr → Ss, with g1f1 = iSs . The idea
of the proof is to find a matrix U ∈ GLr(S) such that UGT
1 =
[
Is
0
]
and
then apply Lemma 1.
We have F T
1 GT
1 = Is and from this we get that the first column g1 of
GT
1 is unimodular, but since r > r − s > sr(S), then g1 is stable, and by
Proposition 7, there exists U1 ∈ Er(S) such that U1g1 = e1. If s = 1, we
finish since GT
1 = g1.
Let s > 2; we have
U1GT
1 =
[
1 ∗
0 F2
]
, F2 ∈ M(r−1)×(s−1)(S).
Note that U1GT
1 has a left inverse (for instance F T
1 U−1
1 ), and the form of
this left inverse is
L =
[
1 ∗
0 L2
]
, L2 ∈ M(s−1)×(r−1)(S),
and hence L2F2 = Is−1. The first column of F2 is unimodular and since
r − 1 > r − s > sr(S) we apply again Proposition 7 and we obtain a
matrix U ′
2 ∈ Er−1(S) such that
U ′
2F2 =
[
1 ∗
0 F3
]
, F3 ∈ M(r−2)×(s−2)(S).
Let
U2 :=
[
1 0
0 U ′
2
]
∈ Er(S),
then we have
U2U1GT
1 =
1 ∗ ∗
0 1 ∗
0 0 F3
.
O. Lezama, C. Gallego 119
By induction on s and multiplying on the left by elementary matrices we
get a matrix U ∈ Er(S) such that
UGT
1 =
[
Is
0
]
.
From this result we get automatically Stafford’s Theorem.
Corollary 4 (Stafford). Let D := An(K) or Bn(K), with char(K) = 0.
Then, any stably free left D-module M satisfying rank(M) > 2 is free.
Proof. The results follows from Theorem 1 since sr(D) = 2.
3. Matrix descriptions of Hermite rings
Rings for which all stably free modules are free have occupied special
attention in homological algebra. In this section we will extend the matrix-
constructive interpretation of commutative Hermite rings given in [15] to
the more general case of noncommutative rings.
Definition 5. Let S be a ring.
(i) S is a projective-free (PF ) ring if every f.g. projective S-module is
free.
(ii) S is a PSF ring if every f.g. projective S-module is stably free.
(iii) S is a Hermite ring, property denoted by H, if any stably free
S-module is free.
The right versions of the above rings (i.e., for right modules) are
defined in a similar way and denoted by PFr, PSFr and Hr, respectively.
We say that S is a PF ring if S is PF and PFr simultaneously; similarly,
we define the properties PSF and H. However, we will prove below later
that these properties are left-right symmetric, i.e., they can be denoted
simply by PF , PSF and H. For domains we will write PFD, PSFD
and HD.
From Definition 5 and Theorem 6 we get that
H ∩ PSF = PF. (4)
The following theorem gives a matrix description of H rings (see [8] and
compare with [15] for the particular case of commutative rings. In [4]
is presented a different and independent proof of this theorem for right
modules).
120 Matrix approach to Hermite rings
Theorem 2. Let S be a ring. Then, the following conditions are equi-
valent.
(i) S is H.
(ii) For every r > 1, any unimodular row matrix u over S of size 1 × r
can be completed to an invertible matrix of GLr(S) adding r − 1
new rows.
(iii) For every r > 1, if u is an unimodular row matrix of size 1 × r,
then there exists a matrix U ∈ GLr(S) such that uU = (1, 0, . . . , 0).
(iv) For every r > 1, given an unimodular matrix F of size s × r, r > s,
there exists U ∈ GLr(S) such that
FU =
[
Is | 0
]
.
Proof. (i) ⇒) (ii): Let u := [u1 · · · ur] and v := [v1 · · · vr]T such that
uv = 1, i.e., u1v1 + · · · + urvr = 1; we define
Sr α−→ S
ei 7→ vi
where {e1, . . . , er} is the canonical basis of the left free module Sr of
columns vectors. Observe that α(uT ) = 1; we define the homomorphism
β : S → Sr by β(1) := uT , then αβ = iS . From this we get that
Sr = Im(β) ⊕ ker(α), β is injective, 〈uT 〉 = Im(β) ∼= S and Im(β) is
free with basis {uT }. This implies that Sr ∼= S ⊕ ker(α), i.e., ker(α) is
stably free of rank r − 1, so by hypothesis, ker(α) is free of dimension
r − 1; let {x1, . . . , xr−1} be a basis of ker(α), then {uT , x1, . . . , xr−1} is
a basis of Sr. This means that
[
uT x1 · · · xr−1
]T
∈ GLr(S), i.e., u can
be completed to an invertible matrix of GLr(S) adding r − 1 rows.
(ii) ⇒) (i): Let M be an stably free S-module, then there exist integers
r, s > 0 such that Sr ∼= Ss ⊕ M . It is enough to prove that M is free for
the case when s = 1. In fact, Sr ∼= Ss ⊕ M = S ⊕ (Ss−1 ⊕ M) is free and
hence Ss−1 ⊕ M is free; repeating this reasoning we conclude that S ⊕ M
is free, so M is free.
Let r > 1 such that Sr ∼= S ⊕ M , let π : Sr −→ S be the canoni-
cal projection with kernel isomorphic to M and let {e1, . . . , er} be the
canonical basis of Sr; there exists µ : S −→ Sr such that πµ = iS
and Sr = ker(π) ⊕ Im(µ). Let µ(1) := uT := [u1 · · · ur]T ∈ Sr, then
π(uT ) = 1 = u1π(e1) + · · · + urπ(er), i.e., v := [π(e1) · · · π(er)]T is such
O. Lezama, C. Gallego 121
that uv = 1, moreover, Sr = ker(π) ⊕ 〈uT 〉. By hypothesis, there exists
U ∈ GLr(S) such that eT
1 U = u.
Let fT : Sr −→ Sr be the homomorphism defined by UT , then
fT (e1) = uT and fT (ei) = ui for i > 2, where u2, . . . , ur are the
others columns of UT (i.e., the transpose of the other rows of U). Since
U = (UT )T then fT is an isomorphism. If we prove that fT (ei) ∈ ker(π)
for each i > 2, then ker(π) is free, and consequently, M is free. In fact, let
f ′ be the restriction of fT to 〈e2, . . . , er〉, i.e., f ′ : 〈e2, . . . , er〉 −→ ker(π).
Then f ′ is bijective: of course f ′ is injective; let w be any vector of Sr, then
there exists x ∈ Sr such that fT (x) = w, we write x := [x1 · · · xr]T =
x1e1 + z, with z = x2e2 + · · · + xrer. We have fT (x) = fT (x1e1 + z) =
x1fT (e1) + fT (z) = x1uT + fT (z) = w. In particular, if w ∈ ker(π), then
w − fT (z) ∈ ker(π) ∩ 〈uT 〉 = 0, so w = fT (z) and hence w = f ′(z), i.e.,
f ′ is surjective.
In order to conclude the proof we will show that fT (ei) ∈ ker(π) for
each i > 2. Since fT was defined by UT , the idea is to change UT in a
such way that its first column was uT and for the others columns were
ui ∈ ker(π), 2 6 i 6 r. Let π(ui) := ri ∈ S, i > 2, and u′
i := ui − riu
T ;
then adding to column i of UT the first column multiplied by −ri we get
a new matrix UT such that its first column is again uT and for the others
we have π(u′
i) = π(ui) − riπ(uT ) = ri − ri = 0, i.e., u′
i ∈ ker(π).
(ii) ⇔ (iii): u can be completed to an invertible matrix of GLr(S) if
and only if there exists V ∈ GLr(S) such that (1, 0, . . . , 0)V = u if and
only if (1, 0, . . . , 0) = uV −1; thus U := V −1.
(iii) ⇒) (iv): The proof will be done by induction on s. For s = 1 the
result is trivial. We assume that (iv) is true for unimodular matrices with
l 6 s − 1 rows. Let F be an unimodular matrix of size s × r, r > s, then
there exists a matrix B such that FB = Is. This implies that the first
row u of F is unimodular; by (iii) there exists U ′ ∈ GLr(S) such that
uU ′ = (1, 0, . . . , 0) = eT
1 , and hence FU ′ = F ′′,
F ′′ =
[
eT
1
F ′
]
,
with F ′ a matrix of size (s − 1) × r. Since FB = Is, then Is = F ′′(U ′−1B),
i.e., F ′′ is an unimodular matrix; let F ′′′ be the matrix obtained elimi-
nating the first column of F ′, then F ′′′ is unimodular of size (s−1)×(r−1),
with r − 1 > s − 1, since the right inverse of F ′′ has the form
[
∗ 0
∗ G′′′
]
.
122 Matrix approach to Hermite rings
By induction, there exists a matrix C ∈ GLr−1(S) such that F ′′′C =
[
Is−1 | 0
]
. From this we get,
FU ′ = F ′′ =
1 0 · · · 0
a′
11 a′
12 · · · a′
1r
...
...
...
a′
s−11 a′
s−12 · · · a′
s−1r
=
[
1 0
∗ F ′′′
]
,
and hence
FU ′
[
1 0
0 C
]
=
[
1 0
∗ F ′′′
] [
1 0
0 C
]
=
[
1 0 0
∗ Is−1 0
]
.
Multiplying the last matrix on the right by elementary matrices we get
(iv).
(iv) ⇒) (iii): Taking s = 1 and F = u in (iv) we get (iii).
From the proof of the previous theorem we get the following result.
Corollary 5. Let S be a ring. Then, S is H if and only if any stably free
S-module M of type Sr ∼= S ⊕ M is free.
Remark 3. (a) If we consider right modules and the right S-module
structure on the module Sr of columns vectors, the conditions of the
previous theorem can be formulated in the following way:
(i)r S is Hr.
(ii)r For every r > 1, any unimodular column matrix v over S of size
r × 1 can be completed to an invertible matrix of GLr(S) adding
r − 1 new columns.
(iii)r For every r > 1, given an unimodular column matrix v over S of
size r × 1 there exists a matrix U ∈ GLr(S) such that Uv = e1.
(iv)r For every r > 1, given an unimodular matrix F of size r × s, r > s,
there exists U ∈ GLr(S) such that
UF =
[
Is
0
]
.
The proof is as in the commutative case, see [15]. Corollary 5 can be
formulated in this case as follows: S is Hr if and only if any stably free
right S-module M of type Sr ∼= S ⊕ M is free.
O. Lezama, C. Gallego 123
(b) Considering again left modules and disposing the matrices of
homomorphisms by rows and composing homomorphisms from the left
to the right (see Remark 1), we can repeat the proof of Theorem 2 and
obtain the equivalence of conditions (i)-(iv). With this notation we do
not need to take transposes in the proof of Theorem 2.
(c) If S is a commutative ring, of course, left and right conditions
are equivalent, see [15]. This follows from the fact that (FG)T = GT F T
for any matrices F ∈ Mr×s(S), G ∈ Ms×r(S). However, as we remarked
before, the Hermite condition is left-right symmetric for general rings
(Proposition 9). Another independent proof of this fact can be found in [4],
Theorem 11.4.4.
4. Matrix characterization of PF rings
In [8] are given some matrix characterizations of projective-free rings;
in this section we present another matrix interpretation of this important
class of rings. The main result presented here (Corollary 7) extends
Theorem 6.2.2 in [15]. This result has been proved independently also
in [4], Proposition 11.4.9.
Theorem 3. Let S be a Hermite ring and M a f.g. projective module
given by the column module of a matrix F ∈ Ms(S), with F T idempotent.
Then, M is free with dim(M) = r if and only if there exists a matrix
U ∈ Ms(S) such that UT ∈ GLs(S) and
(UT )−1F T UT =
[
0 0
0 Ir
]T
. (5)
In such case, a basis of M is given by the last r rows of (UT )−1.
Proof. ⇒): As in the proof of Proposition 1, let f : Ss → Ss be the
homomorphism defined by F and Ss = M ⊕ M ′ with Im(f) = M and
M ′ = ker(f); by the hypothesis M es free with dimension r, so r 6 s (recall
that S is RC). Let h : M → Sr an isomorphism and {z1, . . . , zr} ⊂ M
such that h(zi) = ei, 1 6 i 6 r, then {z1, . . . , zr} is a basis of M . Since S
is an Hermite ring, M ′ is free, let {w1, . . . , ws−r} be a basis of M ′ (recall
that S is IBN ). Then {w1, . . . , ws−r; z1, . . . , zr} is a basis for Ss. With
this we define u in the following way:
u(wj) := ej , for 1 6 j 6 s − r,
u(zi) := es−r+i, for 1 6 i 6 r.
124 Matrix approach to Hermite rings
Note that u is an isomorphism and we get that uf = t0u, where t is given
by t0(ej) := 0 if 1 6 j 6 s − r, and t0(es−r+i) = es−r+i if 1 6 i 6 r; thus,
the matrix of t0 in the canonical basis is
T0 =
[
0 0
0 Ir
]
.
Thus, F T UT = UT T T
0 ; note that (UT )−1 exists since u is an isomorphism,
hence (UT )−1F T UT = T T
0 . From u(zi) := es−r+i we get that (zT
i UT )T =
es−r+i, so zT
i UT = eT
s−r+i and hence zT
i = eT
s−r+i(U
T )−1, i.e., the basis
of M coincides with the last r rows of (UT )−1.
⇐): Let f, u be the homomorphisms defined by F and U , then m(uf) =
m(t0u), where t0 is the homomorphism defined by T0, this means that
uf = t0u, but by the hypothesis UT is invertible, so u is an isomorphism;
from this we conclude that Im(f) ∼= Im(t0), i.e., M = Im(f) ∼= Im(t0) =
〈T0〉 ∼= Sr. Note that this part of the proof does not use that S is an
Hermite ring.
From the previous theorem we get the following matrix description of
PF rings.
Corollary 6. Let S be a ring. S is PF if and only if for each s > 1,
given a matrix F ∈ Ms(S), with F T idempotent, there exists a matrix
U ∈ Ms(S) such that UT ∈ GLs(S) and
(UT )−1F T UT =
[
0 0
0 Ir
]T
, (6)
where r = dim(〈F 〉), 0 6 r 6 s.
Proof. ⇒): Let F ∈ Ms(S), with F T idempotent, and let M be the S-
module generated by the columns of F . By Proposition 1, M is a f.g.
projective module, and by the hypothesis, M is free. Since S is H, we
can apply Theorem 3. If r = dim(M), then r = dim(〈F 〉).
⇐): Let M be a finitely generated projective S-module, so there exists
s > 1 such that Ss = M ⊕M ′; let Ss f−→ Ss be the canonical projection on
M , so F T is idempotent and, by the hypothesis, there exists U ∈ Ms(S)
such that UT ∈ GLs(S) and (6) holds. From the second part of the proof
of Theorem 3 we get that M is free.
Remark 4. (i) If we consider right modules instead of left modules, then
the previous corollary can be reformulated in the following way: S is PFr
O. Lezama, C. Gallego 125
if and only if for each s > 1, given an idempotent matrix F ∈ Ms(S),
there exists a matrix U ∈ GLs(S) such that
UFU−1 =
[
0 0
0 Ir
]
, (7)
where r = dim(〈F 〉), 0 6 r 6 s, and 〈F 〉 represents the right S-module
generated by the columns of F . The proof is as in the commutative case,
see [15].
(ii) Considering again left modules and disposing the matrices of
homomorphisms by rows and composing homomorphisms from the left
to the right (see Remark 1), we can repeat the proofs of Theorem 3 and
Corollary 6 and get the characterization (7) for the PF property; with this
row notation we do not need to take transposes in the proofs. However,
observe that in this case 〈F 〉 represents the left S-module generated by
the rows of F . Note that Corollary 6 could has been formulated this way:
In fact,
[
0 0
0 Ir
]T
=
[
0 0
0 Ir
]
and we can rewrite (6) as (7) changing F T by F (see Remark 1) and
(UT )−1 by U .
(iii) If S is a commutative ring, of course PF = PFr = PF . However,
we will prove in Corollary 8 that the projective-free property is left-right
symmetric for general rings.
Corollary 7. S is PF if and only if for each s > 1, given an idempotent
matrix F ∈ Ms(S), there exists a matrix U ∈ GLs(S) such that
UFU−1 =
[
0 0
0 Ir
]
, (8)
where r = dim(〈F 〉), 0 6 r 6 s, and 〈F 〉 represents the left S-module
generated by the rows of F .
Proof. This is the content of the part (ii) in the previous remark.
Corollary 8. Let S be a ring. S is PF if and only if S is PFr, i.e.,
PF = PFr = PF .
126 Matrix approach to Hermite rings
Proof. Let F ∈ Ms(S) be an idempotent matrix, if S is PF , then there
exists U ∈ GLs(S) such that
UFU−1 =
[
0 0
0 Ir
]
,
where r is the dimension of the left S-module generated by the rows
of F . Observe that UFU−1 is also idempotent, moreover, the matrices
X := UF and Y := U−1 satisfy UFU−1 = XY and F = Y X, then from
Proposition 0.3.1 in [8] we conclude that the left S-module generated
by the rows of UFU−1 coincides with the left S-module generated by
the rows of F , and also, the right S-module generated by the columns
of UFU−1 coincides with the right S-module generated by the columns
of F . This implies that the left S-module generated by the rows of F
coincides with the right S-module generated by the columns of F . Thus,
S is PFr. The symmetry of the problem completes the proof.
Another interesting matrix characterization of PF rings is given in [8],
Proposition 0.4.7: a ring S is PF if and only if given an idempotent
matrix F ∈ Ms(S) there exist matrices X ∈ Ms×r(S), Y ∈ Mr×s(S) such
that F = XY and Y X = Ir. Note that from this characterization we get
again that PF = PFr = PF .
A similar matrix interpretation can be given for PSF rings using
Proposition 0.3.1 in [8] and Corollary 2.
Proposition 8. Let S be a ring. Then,
(i) S is PSF if and only if given an idempotent matrix F ∈ Mr(S)
there exist s > 0 and matrices X ∈ M(r+s)×r(S), Y ∈ Mr×(r+s)(S)
such that
[
F 0
0 Is
]
= XY and Y X = Ir.
(ii) PSF = PSFr = PSF .
Proof. Direct consequence of Proposition 0.3.1 in [8] and Corollary 2.
For the H property we have a similar characterization that proves the
symmetry of this condition.
Proposition 9. Let S be a ring. Then,
(i) S is H if and only if given an idempotent matrix F ∈ Mr(S) with
factorization
O. Lezama, C. Gallego 127
[
F 0
0 1
]
= XY and Y X = Ir, for some matrices
X ∈ M(r+1)×r(S), Y ∈ Mr×(r+1)(S),
there exist matrices X ′ ∈ Mr×(r−1)(S), Y ′ ∈ M(r−1)×r(S) such that
F = X ′Y ′ and Y ′X ′ = Ir−1.
(ii) H = Hr = H.
Proof. Direct consequence of Propositions 0.3.1 and 0.4.7 in [8], and
Corollary 5.
We conclude this section given a matrix constructive proof of a well
known Kaplansky’s theorem.
Proposition 10. Any local ring S is PF .
Proof. Let M a projective left S-module. By Remark 1, part (ii), there
exists an idempotent matrix F = [fij ] ∈ Ms(S) such that the module
generated by the rows of F coincides with M . According to Corollary 7,
we need to show that there exists U ∈ GLs(S) such that the relation (8)
holds. The proof is by induction on s.
s = 1: In this case F = [fij ] = [f ]; since S is local, its idempotents
are trivial, then f = 1 or f = 0 and hence M is free.
s = 2: In view of fact that S is local, two possibilities arise:
f11 is invertible. One can find G ∈ GL2(S) such that GFG−1 =
[
1 0
0 f
]
, for some f ∈ S. In fact, in this case note that
G :=
[
1 f−1
11 f12
−f21f−1
11 1
]
∈ GL2(S) with
G−1 =
[
f11 −f12
f21 −f21f−1
11 f12 + 1
]
.
Since F is idempotent, f so is; applying the case s = 1 we get the result.
1 − f11 is invertible. In the same way, we find H ∈ GL2(S) such that
HFH−1 =
[
0 0
0 g
]
, we take
H :=
[
1 −(1 − f11)−1f12
f21 −f21(1 − f11)−1f12 + 1
]
∈ GL2(S) with
128 Matrix approach to Hermite rings
H−1 =
[
1 − f11 (1 − f11)−1f12
−f21 1
]
.
Note that g is an idempotent of S, then g = 0 or g = 1 and the statement
follows.
Now suppose that the result holds for s − 1; considering both possibil-
ities for f11 we have:
If f11 is invertible, taking
G :=
1 f−1
11 f12 f−1
11 f13 · · · f−1
11 f1s
−f21f−1
11 1 0 · · · 0
−f31f−1
11 0 1 · · · 0
...
...
−fs1f−1
11 0 0 · · · 1
we have that G ∈ GLs(S) and its inverse is:
G−1 =
f11 −f12 −f13 · · · −f1s
f21 −f21f−1
11 f12 + 1 −f21f−1
11 f13 · · · −f21f−1
11 f1s
f31 −f31f−1
11 f12 −f31f−1
11 f13 + 1 · · · −f31f−1
11 f1s
...
...
fs1 −fs1f−1
11 f12 −fs1f−1
11 f13 · · · −fs1f−1
11 f1s + 1
.
Moreover, GFG−1 =
[
1 01,s−1
0s−1,1 F1
]
where F1 ∈ Ms−1(S) is an idempo-
tent matrix. Only remains to apply the induction hypothesis.
If 1 − f11 is invertible, taking
H :=
1 −(1 − f11)−1f12 −(1 − f11)−1f13 · · · −(1 − f11)−1f1s
f21 −f21(1 − f11)−1f12 + 1 −f21(1 − f11)−1f13 · · · −f21(1 − f11)−1f1s
f31 −f31(1 − f11)−1f12 −f31(1 − f11)−1f13 + 1 · · · −f31(1 − f11)−1f1s
.
.
. · · ·
fs1 −fs1(1 − f11)−1f12 −fs1(1 − f11)−1f13 · · · −fs1(1 − f11)−1f1s + 1
we have that H ∈ GLs(S) with inverse given by:
H
−1 =
1 − f11 (1 − f11)−1f12 (1 − f11)−1f13 · · · (1 − f11)−1f1s
−f21 1 0 · · · 0
−f31 0 1 · · · 0
... · · ·
−fs1 0 0 · · · 1
,
and also HFH−1 =
[
0 01,s−1
0s−1,1 F2
]
with F2 ∈ Ms−1(S) an idempotent
matrix. One more time we apply the induction hypothesis.
O. Lezama, C. Gallego 129
5. Some important subclasses of Hermite rings
There are some other classes of rings close related to Hermite rings
(see [8], [11], [12] and [25]) that were studied in [15] for commutative rings,
now we will consider the general noncommutative situation. Some proofs
are easy adaptable from [15] and hence we omit them.
Definition 6. Let S be a ring.
(i) S is an elementary divisor ring (ED) if for any r, s > 1, given a
rectangular matrix F ∈ Mr×s(S) there exist invertible matrices P ∈
GLr(S) and Q ∈ GLs(S) such that PFQ is a Smith normal diagonal
matrix, i.e., there exist d1, d2, . . . , dl ∈ S, with l = min{r, s}, such
that
PFQ = diag(d1, d2, . . . , dl), with Sdi+1S ⊆ Sdi ∩ diS for 1 6 i 6 l,
where SdS denotes the two-sided ideal generated by d.
(ii) S is an ID ring if for any s > 1, given an idempotent matrix
F ∈ Ms(S) there exists an invertible matrix P ∈ GLs(S) such that
PFP −1 is a Smith normal diagonal matrix.
(iii) S is a left K-Hermite ring (KH) if given a, b ∈ S there exist U ∈
GL2(S) and d ∈ S such that U
[
a b
]T
=
[
d 0
]T
. S is a right
K-Hermite ring (KHr) if
[
a b
]
U =
[
d 0
]
. The ring S is KH if
S is KH and KHr.
(iv) S is a left Bézout ring (B) if every f.g. left ideal of S is principal. S
is a right Bézout ring (Br) if every f.g. right ideal of S is principal.
S is a B ring if S is B and Br.
(v) S is a left cancellable ring (C) if for any f.g. projective left S-
modules P, P ′ holds: P ⊕ S ∼= P ′ ⊕ S ⇔ P ∼= P ′. S is right
cancellable (Cr) if for any f.g. projective right S-modules P, P ′
holds: P ⊕ S ∼= P ′ ⊕ S ⇔ P ∼= P ′. S is cancellable (C) if S is (C)
and (Cr).
From Proposition 0.3.1 of [8] it is easy to give a matrix interpretation
of C rings, and also, we can deduce that C = Cr = C.
Proposition 11. Let S be a ring. Then,
(i) S is C if and only if given idempotent matrices F ∈ Ms(S), G ∈
Mr(S) the following statement is true: The matrices
130 Matrix approach to Hermite rings
[
F 0
0 1
]
and
[
G 0
0 1
]
can be factorized as
[
F 0
0 1
]
= X ′Y ′,
[
G 0
0 1
]
= Y ′X ′, for some matrices
X ′ ∈ M(s+1)×(r+1)(S), Y ′ ∈ M(r+1)×(s+1)(S)
if and only if F = XY , G = Y X, for some matrices X ∈ Ms(S),
Y ∈ Mr(S).
(ii) C = Cr = C.
Proof. Direct consequence of Proposition 0.3.1 in [8].
For domains, the above classes of rings are denoted by EDD, IDD,
KHD, KHDr, KHD, BD, BDr, BD and CD, respectively.
Proposition 12. (i) ED ⊆ KH ⊆ B.
(ii) KHD = BD ⊆ PFD.
(iii) PF ⊆ ID
(iv)ID = PF for rings without nontrivial idempotents. In particular,
IDD = PFD.
(v) PF ⊆ C ⊆ H.
Similar relations are valid for KHr, KH, Br and B.
Proof. (i) It is clear that ED ⊆ KH. Let a, b ∈ S, we want to proof that
any left ideal Sa + Sb is principal. There exist U ∈ GL2(S) and d ∈ S
such that U
[
a b
]T
=
[
d 0
]T
, this implies that Sd ⊆ Sa + Sb, but since
[
a b
]T
= U−1
[
d 0
]T
, then Sa + S ⊆ Sd. This proved that KH ⊆ B.
(ii) The equality KHD = BD was proved by Amitsur in [1], Theorem
1.4. The proof of the inclusion BD ⊆ PFD is as in the commutative case,
see [15], Example 6.1.2.
(iii) Using permutation matrices it is clear that PF ⊆ ID (see Corol-
lary 7).
(iv) Let S be an ID ring and let F = [fij ] ∈ Ms(S) be an idempotent
matrix over S; by the hypothesis, there exists P ∈ GLs(S) such that
PFP −1 is diagonal, let D := PFP −1 = diag(d1, d2, . . . , ds); since PFP −1
O. Lezama, C. Gallego 131
is idempotent, then each di is idempotent, so di = 0 or di = 1 for each
1 6 i 6 s. By permutation matrices we can assume that
PFP −1 =
[
0 0
0 Ir
]
,
in addition, note that r is the dimension of the left S-module generated
by the rows of F . Then, S is PF .
(v) Let P, P ′ be f.g. S-modules such that P ⊕S ∼= P ′⊕S; since S is PF
there exists n, n′ such that P ∼= Sn, P ′ ∼= Sn′
and hence Sn ⊕S ∼= Sn′ ⊕S,
so n + 1 = n′ + 1, i.e., P ∼= P ′.
Let now M be a stably free module, M ⊕ Ss ∼= Sr, since r > s and S
is left cancellable, then M ∼= Sr−s.
Remark 5. (a) Z6 shows that PF 6= H and also that PSF 6= H (see [15],
Example 6.1.2). On the other hand, note that if K is a field, then K[x, y] is
PFD but is not BD. Thus, B 6= PF , and consequently, B 6= H, KH 6= H,
ED 6= H. Z[
√
−5] shows that ID 6= H, see [15], Example 6.6.1 and Remark
6.7.14.
(b) In [8] P.M. Cohn asks if there exist examples of H rings that are
not C rings, in other words, C 6= H is probably still an open problem.
(c) It is well known that B 6= Br, a classical example is given by
the skew polynomial ring T [x; σ], where T is a division ring a σ is an
endomorphism of T that is not automorphism. Every left ideal of this
ring is principal, hence, it is a left Bézout ring; but if a /∈ σ(T ), then the
right ideal generated by x and ax is not principal. This example shows
also that KH 6= KHr.
(d) From proposition 12 we conclude that for domains the following
inclusions hold:
EDD ⊆ KHD = BD ⊆ PFD = IDD ⊆ CD ⊆ HD. (9)
Similar relations are valid for the right side.
(e) Many authors have considered additional relations between these
classes of rings in some particular cases, for example, for commutative rings
KH ⊆ H and ID ⊆ H ([12], [16], [23]). In [25] and [26] are analyzed many
cases for which B coincides with KH, see also [10] and [21]. Kaplansky
has conjectured that for commutative domains, BD = EDD (see [11]).
This conjecture probably has not been solved yet.
Next we will see that ID ⊆ H in rings for which all idempotents are
central.
132 Matrix approach to Hermite rings
Proposition 13. Let S be a ring such that all idempotents are central.
Then the following conditions are equivalent:
(i) S is ID.
(ii) Any idempotent matrix over S is similar to a diagonal matrix.
(iii) Given an idempotent matrix F ∈ Mr(S) there exists an unimodular
vector v = [v1, . . . , vr]T over S and an invertible matrix U ∈ GLr(S)
such that Uv = e1 and Fv = av, for some a ∈ S.
Proof. The proof is an easy adaptation of the commutative case, see [15],
Proposition 6.3.2.
Theorem 4. Let S be a ring such that all idempotents are central. Then,
ID ⊆ H.
Proof. We start with the following remark: If u is an unimodular row of
size 1 × r over S and P ∈ GLr(S), then u is completable to an invertible
matrix if and only if uP is completable.
Let u = [u1 · · · ur] be an unimodular row matrix of size 1 × r, there
exists v = [v1 · · · vr]T such that u1v1 + · · · + urvr = 1; we consider the
matrix F = [fij ] ∈ Mr(S), with fij := viuj , 1 6 i, j 6 r. Note that
F 2 = F ; by the hypothesis there exists P ∈ GLr(S) such that PFP −1
is diagonal, let D := PFP −1 = diag(d1, d2, . . . , dr); since PFP −1 is
idempotent, then each di is idempotent. Let w := uP −1 and x := Pv, then
wx = uP −1Pv = 1 and xw = PvuP −1 = PFP −1 = D. By the above
remark, u is completable if and only if w is. Thus, we will show that w is
completable. From xw = D we obtain that xiwi = di is idempotent for all
1 6 i 6 r and xiwj = 0 for i 6= j. But
∑r
k=1 wixi = 1, then wi = wixiwi
and xi = xiwixi. Let fi := wixi for 1 6 i 6 r, hence each fi is idempotent.
By the hypothesis di, fi are central, then di = d2
i = xifiwi = fidi and
fi = f2
i = difi, so that di = fi and xiwi = wixi for 1 6 i 6 r. Therefore,
(
∑r
i=1 xi)(
∑r
i=1 wi) = 1, hence c :=
∑r
i=1 wi is left invertible, c′c = 1.
Observe that cc′ is idempotent, so central, and by the hypothesis there
exists x ∈ S∗ such that xcc′x−1 = d, with d ∈ S idempotent, from this
we get that cc′ = d and c′ = c′d, i.e., c′(1 − d) = 0, so (1 − d)c′ = 0 and
consequently 1 − d = 0, i.e, cc′ = 1. This means that c is invertible. Hence,
the matrix
O. Lezama, C. Gallego 133
V :=
w1 w2 w3 · · · wr
−1 1 0 · · · 0
−1 0 1 · · · 0
...
...
...
...
...
−1 0 0 · · · 1
,
is invertible, i.e., w is completable.
6. Products, quotients and localizations
Next we will study the properties introduced in Definition 6 with respect
to some algebraic standard constructions. Rad(S) represents the Jacobson
radical of the ring S and S∗ the group of units of S.
Proposition 14. Let S be a ring and I ⊆ Rad(S) an ideal of S. Let
{Si}i∈C be a family of rings. Then,
(i) S is H if and only if S/I is H.
(ii)
∏
i∈C Si is H if and only if each Si is H.
(iii) If
∏
i∈C Si is PF , then each Si is PF .
(iv) If S is ED, then S/I is ED for any proper ideal I of S.
(v)
∏
i∈C Si is ED if and only if each Si is ED.
(vi) If S is B, then S/I is B for any proper ideal I of S which is f.g.
as left ideal.
(vii)
∏
i∈C Si is B if and only if each Si is B.
(viii) Suppose that in S all idempotents are central and I is a nilideal. If
S/I is ID, then S is ID.
(ix)
∏
i∈C Si is ID if and only if each Si is ID.
(x) If S is KH, then S/I is KH for any proper ideal I of S.
(xi)
∏
i∈C Si is KH if and only if each Si is KH.
(xii)
∏
i∈C Si is C if and only if each Si is C.
(xiii) If
∏
i∈C Si is PSF , then each Si is PSF .
Similar relations are valid for the right side.
Proof. Some proofs can be adapted from the commutative case (see [15])
or can be get directly from the definition. We include only the proof of
(viii) and (xii): First note that if S := S/I, then U := [uij ] ∈ GLr(S)
134 Matrix approach to Hermite rings
if and only if U = [uij ] ∈ GLr(S). Moreover, let B :=
∏
i∈C Si, then
Ms(B) ∼= ∏
i∈C Ms(Si), where the isomorphism is defined by F 7→ (F (i)),
with F = [fuv], fuv = (f
(i)
uv ), F (i) = [f
(i)
uv ]. From this we obtain that
Ms(B)∗ = GLs(B) ∼= ∏
i∈C GLs(Si) =
∏
i∈C Ms(Si)
∗.
(viii) Let F ∈ Ms(S) be an idempotent matrix, then F ∈ Ms(S) is
idempotent and there exists P ∈ GLs(S) such that
D = P F (P )−1 = diag(d1, . . . , dr), with S di+1 S ⊆ S di ∩ di S.
Note that D is idempotent, so each di is idempotent, 1 6 i 6 r; let
d := d1 · · · dr, then d
2
= d. Since I is nilideal we can assume that d is
idempotent (see [13]), and hence, central; moreover since each di is central,
di|di+1, and then d = dr (this can be easy prove by induction on r). Note
that Der = der, so Fv = dv, with v := (P )−1er unimodular over S,
and hence, v is unimodular over S. Moreover, there exists V ∈ GLr(S)
such that V v = e1. In fact, we have v − P −1er = u = [u1, . . . , ur]T , with
ui ∈ Rad(S), 1 6 i 6 r. Then, v = P −1er + u, and hence, Pv = er + Pu
is a column matrix with the last component invertible, so multiplying
by elementary and permutation matrices we get V ∈ GLr(S) such that
V v = e1.
We have Fv = dv + z, with z = [z1, . . . , zr]T , zi ∈ Rad(S), 1 6 i 6 r.
From this we get that F 2v = Fv = dFv + Fz, so Fz = (1 − d)Fv =
(1 − d)(dv + z) = (1 − d)z since (1 − d)d = 0. Then, F (v + (2d − 1)z)
= Fv+(2d−1)Fz = dv+z +(2d−1)(1−d)z = dv+dz = d(v+(2d−1)z).
Thus, given the idempotent matrix F we have found a vector w :=
v + (2d − 1)z and an element d ∈ S such that Fw = dw, moreover w is
unimodular since v is unimodular and zi ∈ Rad(S), 1 6 i 6 r. In addition,
the first component of the vector V w = e1 + V (2d − 1)z is invertible,
so by elementary operations we found a matrix W ∈ GLr(S) such that
Ww = e1. From Proposition 13 we get that S is an ID ring.
(xii) ⇒): We will apply Proposition 11. Let k ∈ C and F (k) = [f
(k)
uv ] ∈
Ms(Sk), G(k) = [g
(k)
uv ] ∈ Mr(Sk) idempotent matrices, then F ∈ Ms(B),
G ∈ Mr(B) are idempotent, where F = [fuv], G = [guv], with fuv =
(f
(i)
uv ), guv = (g
(i)
uv ) and f
(i)
uv = 0 = g
(i)
uv for i 6= k. Since B is a C ring, the
enlarged matrices
[
F 0
0 1
]
and
[
G 0
0 1
]
can be factorized as in Proposition 11 if and only if the matrices F, G can
be factorized. This implies that the matrices
O. Lezama, C. Gallego 135
[
F (k) 0
0 1
]
and
[
G(k) 0
0 1
]
can be factorized if and only if the matrices F (k), G(k) can be factorized.
This proves that Sk is a C ring.
⇐): Let F = [fuv] ∈ Ms(B), G = [guv] ∈ Mr(B) be idempotent
matrices, with fuv = (f
(k)
uv ), guv = (g
(k)
uv ), f
(k)
uv , g
(k)
uv ∈ Sk; since each ring
Sk is C, we can repeat the previous reasoning, but in the inverse order,
and conclude that B is a C ring.
Now we will consider the localizations of some classes of rings intro-
duced in Definition 6.
Proposition 15. Let S be a ring and T a multiplicative system of S
such that T −1S exits. If S is ED (KH, B), then T −1S is ED (KH, B).
Similar properties are valid for the right side.
Proof. Let S a ED ring and F ∈ Mr×s(T
−1S), then F = [fij ] with
fij = t−1
ij sij , where tij ∈ T and sij ∈ S, for 1 6 i 6 r, 1 6 j 6 s. By
Proposition 2.1.16 in [17], there exist t ∈ T and lij ∈ S such that fij =
t−1lij , then tF = [lij ] ∈ Mr×s(S), hence tF admits a diagonal reduction,
i.e., there exist P ∈ GLr(S) and Q ∈ GLs(S) such that P (tF )Q =
diag(d1, . . . , dl), with d1, . . . , dl ∈ S, l = min{r, s} and Sdi+1S ⊆ Sdi∩diS.
Note that Pt, Q ∈ GLr(T −1S). Thus, (Pt)FQ = P (tF )Q = D, moreover,
T −1Sdi+1T −1S ⊆ T −1Sdi ∩ diT
−1S. This proves that T −1S is ED.
The proof for KH is completely analogous.
Suppose now that S is a B ring and let J be a f.g. left ideal of
T −1S, then J = 〈q1, . . . , qr} where qi = t−1
i si with ti ∈ T and si ∈ S
for 1 6 i 6 r. Let t ∈ T and ai ∈ S such that qi = t−1qi, then tqi = ai.
Therefore, J ′ := T −1S a1
1 + · · · + T −1S ar
1 ⊆ J ; but J ⊆ J ′ : in fact,
let x = b1
t1
q1 + · · · + br
tr
qr ∈ J , then x = t−1
1 b1t−1 a1
1 + · · · + t−1
r brt−1 ar
1 ;
since bit
−1 ∈ T −1S exist, b′
i ∈ S and li ∈ T such that bit
−1 = l−1
i b′
i,
1 6 i 6 r, hence x = t−1
1 l−1
1 b′
1
a1
1 + · · · + t−1
r l−1
r b′
r
ar
1 = (l1t1)−1b′
1
a1
1 + · · · +
(lrtr)−1b′
r
ar
1 ∈ J ′. Thus, J = J ′.
Now note that J ′ = T −1I, where I := Sa1 + · · · + Sar: clearly
T −1I ⊆ J ′; let y ∈ J ′, then y = b1
s1
a1
1 + · · · + br
sr
ar
1 = b1a1
s1
+ · · · + brar
sr
=
c1b1a1+···+crbrar
u
for some ci ∈ S and u ∈ T . Hence y = u−1(c1b1a1 + · · · +
crbrar) ∈ T −1I. But I is a f.g. left ideal of S, then I = 〈a} for some
a ∈ S, and therefore J = T −1S a
1 , i.e., J is principal.
136 Matrix approach to Hermite rings
We observe that if S is B (or KH) and T a multiplicative system of S
such that T −1S and ST −1 exist, then T −1S is B (KH) since ST −1 ∼= T −1S.
On the other hand, if S is H (PF , PSF) not always T −1S has the
correspondent property (see [8]).
For the localization by primes ideals we need to recall a definition.
Let S be a left Noetherian ring and P a prime ideal of S. It says that P
is left localizable if the set
S(P ) := {a ∈ S|a ∈ S/P is not a zero divisor}
is a multiplicative system of S and S(P )−1S exists; we will write SP :=
S(P )−1S. Right localizable prime ideals are defined similarly (see [3]).
Theorem 5. Let S be a left Noetherian ring.
(i) If P is a left (right) localizable prime ideal, then SP is H.
(ii) If P is a left (right) localizable completely prime ideal, then SP is
PF , and hence, C and PSF .
Proof. (i) It is well known ([3]) that Sp has a unique maximal ideal
PSP := {a
s
| a ∈ P, s ∈ S(P )}; moreover, Rad(SP ) = PSP and Sp/PSp
∼=
Ql(S/P ) is simple Artinian, where Ql(S/P ) denotes the left quotient ring
of S/P . Therefore, SP is a semilocal ring and hence SP is H (from
Theorem 1 follows that any left Artinian ring S is H since sr(S) = 1, so
semilocal rings are H).
(ii) If P is completely prime, S/P is a domain, so that Ql(S/P ) is a
division ring, and therefore, SP is a local ring. From Proposition 10 we
get that SP is PF ⊆ C ∩ PSF .
References
[1] S.A. Amitsur, Remarks of principal ideal rings, Osaka Math. Journ. 15, (1963),
59-69.
[2] H. Bass, Algebraic K-theory, Benjamin, 1968.
[3] A. Bell, Notes on Localizations in Noncommutative Noetherian Rings, Departa-
mento de Álgebra y Fundamentos, Universidad de Granada, España, 1989.
[4] H. Chen, Rings Related to Stable Range Conditions, World Scientific: Series in
Algebra, Vol. 11, 2011.
[5] F. Chyzak, A. Quadrat, and D. Robertz, Effective algorithms for parametrizing
linear control systems over Ore algebras, Appl. Algebra Engrg. Comm. Comput.,
16, 2005, 319-376.
[6] T. Cluzeau and A. Quadrat, Factoring and decomposing a class of linear functional
systems, Lin. Alg. And Its Appl., 428, 2008, 324-381.
O. Lezama, C. Gallego 137
[7] T. Cluzeau and A. Quadrat, Serre’s reduction of linear partial differential systems
with holonomic adjoints, J. of Symb. Comp., 47, 2012, 1192-1213.
[8] P. Cohn, Free Ideal Rings and Localizations in General Rings, Cambridge Uni-
versity Press, 2006.
[9] K. Goodearl and R. Jr. Warfield, An Introduction to Noncommutative Noetherian
Rings, London Mathematical Society, ST 61, 2004.
[10] A.I. Hatalevych, Right Bézout rings with waist is a right Hermite ring, Ukr. Math.
J., 62, 2010, 151-154.
[11] I. Kaplansky, Elementary divisors and modules, Trans. Amer. Math. Soc. 66
(1949), 464-491.
[12] T.Y. Lam, Serre’s Problem on Projective Modules , Springer Monographs in
Mathematics, Springer, 2006.
[13] J. Lambeck, Lectures on Rings and Modulea, Chelsea Publishing Company, 1986.
[14] O. Lezama et. al., Quillen-Suslin rings, Extracta Mathematicae, 24, 2009, 55-97.
[15] O. Lezama, Matrix and Gröbner Methods in Homological Algebra over Commu-
tative Polynomial Rings, Lambert Academic Publishing, 2011.
[16] B. MacDonald, Linear Algebra over Commutative Rings, Marcel Dekker, 1984.
[17] J. McConnell and J. Robson, Noncommutative Noetherian Rings, Graduate
Studies in Mathematics, AMS, 2001.
[18] A. Quadrat and D. Robertz, Computation of bases of free modules over the Weyl
algebras, J. Symb. Comp., 42, 2007, 1113-1141.
[19] A. Quadrat, Internal stabilization of coherent control systems, IFAC, 2001.
[20] A. Quadrat, “Stabilizing” the stabilizing controllers, preprint.
[21] O. M. Romaniv, Elementary reduction of matrices over right 2-Euclidean rings,
Ukr. Math. J., 56, 2004, 2028-2034.
[22] J.T. Stafford, Module structure of Weyl algebras, J. London Math. Soc. 18, 1978,
429-442.
[23] A. Steger, Diagonability of idempotent matrices, Pac. J. Math., 19, 1966, 535-542.
[24] C. A. Weibel, Algebraic K-theory, preprint, 2012.
[25] B. Zabavsky, Diagonalizability theorems for matrices over rings with finite stable
range, Algebra and Discrete Mathematics, 1, (2005), 151-165.
[26] B. Zabavsky, Reduction of matrices with stable range not exceeding 2, Ukr. Math.
Zh. (in Ucraine), 55, (2003), 550-554.
Contact information
O. Lezama,
C. Gallego
Departamento de Matemáticas, Universidad Na-
cional de Colombia, Ciudad Universitaria, Bo-
gotá, Colombia.
E-Mail: jolezamas@unal.edu.co,
cmgallegoj@unal.edu.co
URL: www.matematicas.unal.edu.co/sac2
Received by the editors: 22.12.2012
and in final form 12.03.2013.
|