Steadiness of polynomial rings
A module M is said to be small if the functor Hom(M,−) commutes with direct sums and right steady rings are exactly those rings whose small modules are necessary finitely generated. We give several results on steadiness of polynomial rings, namely we prove that polynomials over a right perfect ring...
Gespeichert in:
| Datum: | 2010 |
|---|---|
| 1. Verfasser: | |
| Format: | Artikel |
| Sprache: | English |
| Veröffentlicht: |
Інститут прикладної математики і механіки НАН України
2010
|
| Schriftenreihe: | Algebra and Discrete Mathematics |
| Online Zugang: | http://dspace.nbuv.gov.ua/handle/123456789/154871 |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Zitieren: | Steadiness of polynomial rings / J. Zemlicka // Algebra and Discrete Mathematics. — 2010. — Vol. 10, № 2. — С. 107–117. — Бібліогр.: 13 назв. — англ. |
Institution
Digital Library of Periodicals of National Academy of Sciences of Ukraine| id |
irk-123456789-154871 |
|---|---|
| record_format |
dspace |
| spelling |
irk-123456789-1548712019-06-17T01:31:16Z Steadiness of polynomial rings Zemlicka, J. A module M is said to be small if the functor Hom(M,−) commutes with direct sums and right steady rings are exactly those rings whose small modules are necessary finitely generated. We give several results on steadiness of polynomial rings, namely we prove that polynomials over a right perfect ring such that EndR(S) is finitely generated over its center for every simple module S form a right steady ring iff the set of variables is countable. Moreover, every polynomial ring in uncountably many variables is non-steady. 2010 Article Steadiness of polynomial rings / J. Zemlicka // Algebra and Discrete Mathematics. — 2010. — Vol. 10, № 2. — С. 107–117. — Бібліогр.: 13 назв. — англ. 2000 Mathematics Subject Classification:16S36, 16D10. http://dspace.nbuv.gov.ua/handle/123456789/154871 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 |
A module M is said to be small if the functor Hom(M,−) commutes with direct sums and right steady rings are exactly those rings whose small modules are necessary finitely generated. We give several results on steadiness of polynomial rings, namely we prove that polynomials over a right perfect ring such that EndR(S) is finitely generated over its center for every simple module S form a right steady ring iff the set of variables is countable. Moreover, every polynomial ring in uncountably many variables is non-steady. |
| format |
Article |
| author |
Zemlicka, J. |
| spellingShingle |
Zemlicka, J. Steadiness of polynomial rings Algebra and Discrete Mathematics |
| author_facet |
Zemlicka, J. |
| author_sort |
Zemlicka, J. |
| title |
Steadiness of polynomial rings |
| title_short |
Steadiness of polynomial rings |
| title_full |
Steadiness of polynomial rings |
| title_fullStr |
Steadiness of polynomial rings |
| title_full_unstemmed |
Steadiness of polynomial rings |
| title_sort |
steadiness of polynomial rings |
| publisher |
Інститут прикладної математики і механіки НАН України |
| publishDate |
2010 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/154871 |
| citation_txt |
Steadiness of polynomial rings / J. Zemlicka // Algebra and Discrete Mathematics. — 2010. — Vol. 10, № 2. — С. 107–117. — Бібліогр.: 13 назв. — англ. |
| series |
Algebra and Discrete Mathematics |
| work_keys_str_mv |
AT zemlickaj steadinessofpolynomialrings |
| first_indexed |
2025-07-14T06:56:20Z |
| last_indexed |
2025-07-14T06:56:20Z |
| _version_ |
1837604464405512192 |
| fulltext |
Algebra and Discrete Mathematics RESEARCH ARTICLE
Volume 10 (2010). Number 2. pp. 107 – 117
c© Journal “Algebra and Discrete Mathematics”
Steadiness of polynomial rings
Jan Žemlička
Communicated by V. V. Kirichenko
Abstract. A module M is said to be small if the functor
Hom(M,−) commutes with direct sums and right steady rings
are exactly those rings whose small modules are necessary finitely
generated. We give several results on steadiness of polynomial rings,
namely we prove that polynomials over a right perfect ring such
that EndR(S) is finitely generated over its center for every simple
module S form a right steady ring iff the set of variables is countable.
Moreover, every polynomial ring in uncountably many variables is
non-steady.
The notion of a small module is one of the most natural variant of
a concept of compactness in categories of modules. It is defined as a
module M for which the covariant functor Hom(M,−) commutes with all
direct sums of modules, which exactly means that every homomorphism
of M into an arbitrary direct sum
⊕
i∈I Ni of modules can be factorized
through a suitable direct sum
⊕
i∈F Ni where F ⊆ I is finite. The first
systematic research concerning this notion was published in [8], however
a non-categorical characterization as well as an observation that there
are examples of infinitely generated small modules had appeared in [2,
p.54]. Small modules are studied under various terms (module of type
Σ, dually slender, U-compact) and motivation of the research have come
in particular from the theory of representable equivalences of module
categories ([4], [5], [6] [9], [10] etc.) and the structure theory of graded
rings [7] and almost free modules [11].
It is easy to see (and it will be obvious from Lemma 1.1) that every small
module is finitely generated. As many examples of infinitely generated
This work is part of the research project MSM 0021620839, financed by MŠMT.
2000 Mathematics Subject Classification: 16S36, 16D10.
Key words and phrases: small module, steady ring, polynomial ring.
108 Steadiness of polynomial rings
small modules are known it seems to be useful to define a right steady
ring as a ring over which classes of all small modules and of all finitely
generated modules coincide. However any general ring-theoretical criterion
of right steadiness is not available, there are known large (and frequently
studied) classes of right steady rings (right noetherian, right perfect, right
semiartinian with countable socle length) as well as right non-steady rings
(infinite products of rings, endomorphism rings of infinitely generated free
module, simple rings of infinite right rank). Moreover, characterization
of right steady rings is known in some special cases (for commutative
semiartinian, serial, and self injective regular rings) and a module-theoretic
criterion of steadiness via products of simple modules end their injective
envelops is proved in [13].
The present paper focuses on the question, how polynomial rings
with commuting variables reflect steadiness. Obviously, polynomials over
a right non-steady ring form a right non-steady ring. As every right
noetherian ring is right steady by [8, 7o], right steadiness of polynomials
over a noetherian ring in finitely many variables follows from Hilbert basis
theorem. Moreover, Rentschler proved in the cited work using the classical
commutative prime-ideal calculus that polynomial rings in countably many
variables over commutative noetherian rings are steady. Nevertheless, the
existence of infinitely generated small modules over R[X] for a general
right steady ring R remains to be an open problem not only for countable
but even for finite X. Our main result extends the Rentschler’s quoted
theorem to a "less commutative" case; namely, we prove that polynomial
ring T [X] is right steady, if X is countable and T is a skew field finitely
generated over its center (Theorem 2.1). It implies right steadiness of
polynomials R[X] in countably many variables whenever R is a right
perfect such that EndR(S) is finitely generated right module over its
center for every simple S (Theorem 2.7). On the other hand, we show
that polynomials in uncountably many variables forms non-steady ring in
general (Proposition 3.2).
Throughout the paper a ring R means an associative ring with unit,
and a module means a right R-module. We say that N is a subfactor of
M if it is a submodule of a suitable factor of M . A submodule N of M
is called superfluous if N + X 6= M for every proper submodule X of
M (note that we will use the term small exclusively in the sense defined
above). We denote by J(R) Jacobson radical and by Z(R) the center of
R. The symbols ω and ω1 respectively means the first infinite and the
first uncountable ordinal respectively. Note that we identify cardinals with
the least ordinals of the corresponding cardinality and "countable" means
finite or infinitely countable.
For non-explained terminology we refer to [1].
J. Žemlička 109
1. Preliminaries
We start with well-known characterization of small modules (see e.g. [9,
Lemma 1.2], [5, Lemma 1.1] and [8, 1o]).
Lemma 1.1. The following conditions are equivalent for an arbitrary
module M:
(1) M is small,
(2) if M =
⋃
i<ω Mi for an increasing chain of submodules Mi ⊆ Mi+1 ⊆
M , then there exists n such that M = Mn,
(3) if M =
∑
i<ω Mi for a system of submodules Mi ⊆ M , then there
exists n such that M =
∑
i<nMi.
We will use freely several easy consequences of the previous lemma:
Corollary 1.2. Let M be a small module.
(1) Any factor of M is small,
(2) if M is countable generated, M is finitely generated,
(3) if M =
⊕
i∈I Mi, then I is finite,
(4) if M is a submodule of
∑
i<ω Ni, there exists n such that M ⊆∑
i<nNi,
(5) if Mi are submodules of M such that M/
∑
i<nMi is infinitely gen-
erated for each n, M/
∑
i<ω Mi is infinitely generated.
Before we apply the fact that no infinitely generated small module is
countable generated we make several technical observations about count-
ably generated modules and ideals. First one is a "countable" analogue of
Hilbert basis theorem.
Lemma 1.3. Let R be a ring whose all right ideals are countably generated.
(1) Every submodule of every countably generated module is countably
generated.
(2) If X is a countable set of variables, every right ideal of the polyno-
mial ring R[X] is countably generated.
Proof. (1) Fix a countable set {m0,m1, . . . } of generators of M . Put
M0 = 0, Mn =
∑
i<nmiR for n > 0 and assume that N is an uncountably
generated submodule of M . Since N =
⋃
n<ω(N ∩Mn), there exists n such
that N∩Mn is uncountably generated; take such a minimal n. As N∩Mn−1
110 Steadiness of polynomial rings
is countably generated, (N ∩Mn)/(N ∩Mn−1) is uncountable generated.
Finally,Mn/Mn−1 is a cyclic module and ((N∩Mn)+Mn−1)/Mn−1
∼= (N∩
Mn)/(N ∩Mn−1) is its uncountable generated submodule, a contradiction
with the hypothesis.
(2) Since R[X] is countably generated as a right R-module, every
its R-submodule is countably generated by (1), hence every right R[X]-
submodule of R[X] is countable generated as well.
The following technical lemma, which generalizes [12, Lemma 6], uses
the similar argument as Lemma 1.3.
Lemma 1.4. Let R be a ring, M a module and M =
∑
i<ω Mi where
Mi are submodules of M such that no subfactor of Mi is an infinitely
generated small module. Then no subfactor of M is an infinitely generated
small module.
Proof. Since every small submodule of any factor module M/X is a
submodule of
∑
i≤n(Mi +X/X) for a suitable n by Corollary 1.2(4), it is
enough to show that there exists no infinitely generated small subfactor
of
∑
i≤nMi.
Assume that there exists an infinitely generated small submodule N
of some factor
∑
i≤nMi/Y , fix a minimal such n. We may suppose that
Y = 0. Since N +
∑
i<nMi/
∑
i<nMi is a submodule of the module∑
i≤nMi/
∑
i<nMi
∼= Mn/(Mn ∩
∑
i<nMi), it is finitely generated be-
cause no small subfactor of Mn is infinitely generated. Hence there exists
a finitely generated module such that F +
∑
i<nMi = N +
∑
i<nMi. Now,
N/F is an infinitely generated small submodule of
∑
i<nMi+F/F , which
is a contradiction with the minimality of n.
Lemma 1.5. Let R be a ring whose all right ideals are countably generated,
M an infinitely generated small module, and κ an infinite cardinal. Suppose
that
⊕
α<κNα is a submodule of M such that M/(
⊕
α∈K Nα) is infinitely
generated for each finite subset K ⊂ κ. Then M/(
⊕
α<κNα) is an infinitely
generated module.
Proof. The assertion for countable cardinals follows immediately from
Corollary 1.2(5).
Let κ be uncountable and assume that there exists a finitely generated
module F such that M = F +
⊕
α<κNα. Put NU =
⊕
α∈U Nα for a
arbitrary subset U ⊂ κ and N =
⊕
α<κNα. By the hypothesis and by
Lemma 1.3(1) all submodules of F are countably generated, hence there
exists a countable set C ⊂ κ such that F ∩N ⊆ NC , which implies that
(F + NC) ∩ Nκ\C = 0. Hence M/(F + NC) = (F + N)/(F + NC) ∼=
J. Žemlička 111
Nκ\C/((F +NC) ∩Nκ\C) ∼= Nκ\C . But M is small, a contradiction with
Corollary 1.2(3).
Recall several well-known properties of the class of all right steady
rings.
Proposition 1.6. Let R be a right steady ring and Ri a ring for each
i ∈ I.
(1) Every factor of R is right steady,
(2) every ring Morita equivalent to R is right steady,
(3)
∏
i∈I Ri is right steady iff I is finite and each Ri is right steady.
Proof. (1) [5, Lemma 1.9], (2) [10, Theorem 2.5], (3) [6, Lemma 1.7].
2. Polynomials in countably many variables
Let R be a ring, M a module and r ∈ R. We say that M is r-torsion-free
provided mr 6= 0 for each nonzero m ∈ M . If S ⊂ R we say that M is
S-torsion-free if M is r-torsion-free for each r ∈ S. Denote by M(X) a
set of all monic monomials of the polynomial ring R[X].
Theorem 2.1. Let T be a skew field finitely generated over its center
Z(T ) and X be a countable set of variables. Then the polynomial ring
T [X] is right steady.
Proof. Assume that T [X] is not right steady, i.e. there exists an infinitely
generated small right T [X]-module. As all ideals of T [X] are countably
generated by Lemma 1.3(2) we may apply [12, Lemma 11] which says that
there exist a two-sided prime ideal I and a module M such that
(+) M is infinitely generated and small, MI = 0 and M/MpT [X] is
finitely generated for every p ∈ T [X] \ I.
It is easy to see that every infinitely generated factor of M satisfies
the condition (+) as well.
Before we finish the proof of Theorem 2.1, we prove three technical
lemmas, in which we will deal with one fixed ideal I for some modules M
satisfying the condition (+).
For convenience we introduce some new notation. Put C = Z(T ) and
S = Z(T )[X] \ I. Obviously, C is a field and S is a multiplicative set of
the ring T [X]. Finally, put Lp = {m ∈ L; mp = 0} for every p ∈ S and
every module L. Clearly, Lp is a module because p is a central polynomial.
112 Steadiness of polynomial rings
Lemma 2.2. Assume that a module M satisfies the condition (+) and
let Y = {y0, . . . , yn} ⊂ M(X). Then there exists a submodule MY ⊆ M
such that M/MY is an infinitely generated small module and M/MY is
s-torsion-free for each polynomial of the form s =
∑
i<n aiyi ∈ S.
Proof. We will prove the assertion by induction on the cardinality of the
set Y . If Y = ∅, the claim is true for MY = 0. Suppose that the assertion
holds true for all modules M satisfying the condition (+) and for every
Y such that card(Y ) ≤ n. Let card(Y ) = n+ 1 where Y = {y0, . . . , yn}.
Define two sets U = {
∑
i≤n aiyi ∈ S} and V = {y0, . . . , yn−1}.
We define inductively two chains {Pi}i<ω and {Ni}i<ω of submodules
of M such that Pi−1 ⊆ Ni−1 ⊆ Pi, Pi/Ni−1 = (M/Ni−1)V and M/Ni is
infinitely generated. Put P0 = N0 = 0. Suppose that Pi−1 and Ni−1 are
defined and note that there exists module (M/Ni−1)V by the induction
hypothesis since M/Ni−1 is infinitely generated and it satisfies (+). Hence
we may define Pi as the submodule of M containing Ni−1 for which
Pi/Ni−1 = (M/Ni−1)V and Ni the submodule of M containing Pi such
that Ni/Pi =
∑
p∈U (M/Pi)p. Note that M/Pi
∼= (M/Ni−1)/(M/Ni−1)V
is infinitely generated by the induction premise. It remains to prove that
M/Ni is infinitely generated as well.
Put M = M/Pi and fix an arbitrary p ∈ S. Since p is a central element,
it acts on M as an endomorphism; denote by ̺p the endomorphism induced
by the multiplication by p. Then Mp = ker̺p and M/Mp is a finitely
generated module by (+). Hence M/Mp
∼= ̺p(M) = Mp is infinitely
generated.
Now, fix p, q ∈ S such that p =
∑
i≤n aiyi, q =
∑
i≤n biyi are C-linear
combinations of monomials from Y . Suppose that m ∈ Mp ∩ M q is a
nonzero element. As M = M/Pi
∼= (M/Ni−1)/(M/Ni−1)V , polynomials p
and q are not C-linear combinations of monomials from V , i.e. both an
and bn are nonzero. Moreover, m(p− qb−1
n an) = 0 because mp = mq = 0
and p− qb−1
n an is a C-linear combination of elements from the set V . It
implies that (p − qb−1
n an) ∈ I by the induction hypothesis and because
M is not (p − qb−1
n an)-torsion-free. We have proved that (p − qb−1
n an)
annihilates M . Hence mp = 0 iff mqb−1
n an = 0 and, obviously, it holds
true iff mq = 0, or equivalently formulated Mp = M qb−1
n an
= M q.
We define an equivalence ∼ on U = {
∑
i≤n aiyi ∈ S}. For p, q ∈ U we
have p ∼ q provided there exists a non-zero a ∈ C such that p− qa ∈ I.
We have proved that p 6∼ q iff Mp ∩ M q = 0, and p ∼ q iff Mp = M q.
Moreover,
∑
p∈U Mp =
⊕
[p]∈U/∼Mp. Indeed, suppose that
∑k
i=0mi = 0
where 0 6= mi ∈ Mpi , pi ∈ S, pi 6∼ pj for every i 6= j and fix the minimal
positive k satisfying this condition. Then 0 = m0p0 = −
∑k
i=1mip0, hence
J. Žemlička 113
by minimality of k it holds true that mip0 = 0 for each i > 0. Since it
implies that mi ∈ Mp0 , we obtain a contradiction.
Finally, we are ready to show that the hypothesis of Lemma 1.5
is satisfied for the submodule
⊕
[p]∈U/∼Mp of the module M . Fix an
arbitrary finite set of polynomials p1, . . . , pr ∈ S and put p = p1 · . . . ·
pr ∈ S. Since
∑r
i=1Mpi ⊆ Mp and since Mp ∼= M/Mp is an infinitely
generated homomorphic image of M/
∑r
i=1Mpi , we see that M/
∑r
i=1Mpi
is infinitely generated as well. We may apply Lemma 1.5 which says that
(M/Ni) ∼= M/
∑
p∈U Mp= M/
⊕
[p]∼∈U/∼Mp is an infinitely generated
module.
Remind that we have constructed chains {Pi}i<ω and {Ni}i<ω of
submodules of M such that Pi−1 ⊆ Ni−1 ⊆ Pi, Pi/Ni−1 = (M/Ni−1)V
and M/Ni is infinitely generated. Applying Corollary 1.2(5) we see that
M/
⋃
i<ω Pi = M/
⋃
i<ω Ni is an infinitely generated small module. Finally,
fix p ∈ U and mp ∈
⋃
i<ω Ni. Then there exists k such that mp ∈ Nk ⊆
Pk+1 thus m ∈ Nk+1, which implies that M/
⋃
i<ω Ni is U -torsion-free
over. Thus we may put MY =
⋃
i<ω Ni.
Lemma 2.3. If a module M satisfies the condition (+), there exists an
infinitely generated factor of M which is S-torsion-free.
Proof. Fix an increasing chain (Yi; i < ω) of finite subsets Yi ⊂ Yi+1 ⊂
M(X) such that
⋃
i<ω Yi = M(X). Then applying Lemma 2.2 we con-
struct a countable chain of submodules of M such that M0 = 0 and
Mi/Mi−1 = (M/Mi−1)Yi
. Obviously, M/Mi
∼= (M/Mi−1)/(M/Mi−1)Yi
is infinitely generated for each i > 0, hence M = M/(
⋃
i<ω Mi) is an
infinitely generated small module by Corollary 1.2(5). It is easy to see
that M is S-torsion-free.
Recall that a ring is semisimple if it is (direct) sum of its simple
submodules, or, equivalently, if it is isomorphic to a finite direct product
of matrix rings over skew fields.
Lemma 2.4. If a small module M is S-torsion-free and MI = 0, then
M is finitely generated.
Proof. By [3, Proposition 0.5.3] there exists a right ring of fractions R of
the ring T [X]/I with respect to the multiplicative set ((C[X] + I/I) \ {I}.
Note that there exists the natural embedding of T [X]/I into R since the
multiplicative set is contained in the center of T [X]/I.
First, we prove that R is a semisimple ring. Denote by Q the classical
ring of fractions of C[X]/I. Obviously, Q is a field which is a subring of
R and R is a finitely generated vector space over Q by the hypothesis of
Theorem 2.1. Since each right ideal in R is a Q-subspace of R, every strictly
114 Steadiness of polynomial rings
decreasing chain of right ideals is finite, hence R is right artinian. Since I
is a prime ideal, no nonzero ideal of T [X]/I and so of R is nilpotent, hence
J(R) = 0, which implies that R = R/J(R) is semisimple by Hopkins
theorem.
Now, we show that no factor of R contains as a right T [X]/I-module
an infinitely generated small submodule. Assume to contrary that some
factor of R contains such a submodule. Since T [X]/I ⊆ R, there exists an
infinitely generated small submodule M of a suitable factor of R/(T [X]/I),
otherwise some factor of cyclic module T [X]/I contains an infinitely
generated (so uncountably generated) small module which contradicts to
Lemma 1.3. Obviously any subfactor of R/(T [X]/I) is torsion (i.e. for
every m ∈ M there exists nonzero r ∈ T [X]/I such that mr = 0). As
M has the natural structure of (infinitely generated small) T [X]-module
such that MI = 0, we may use [12, Lemma 11], which claims that there
exists a factor M of M and a nontrivial ideal J containing I such that
the condition (+) holds true for M . Applying Lemma 2.3 we obtain a
S′-torsion-free factor of M where S′ = C[X] \ J , which is a contradiction
because M should be torsion.
Finally, there exists an embedding M →֒ M ⊗T [X]/I R ∼=R
⊕
α<κ Sα
by [3, Proposition 0.6.1], where κ is a cardinal and every Sα, α < κ, is a
simple right R-module, since R is semisimple. Note that we have proved
that no T [X]/I-factor of any Sα contains an infinitely generated small
submodule. Since M is small , there exists n < ω and αi < κ, for each
i < n, such that M →֒
⊕
i<k Sαi
. As no factor of Sα, for each α < κ,
contains an infinitely generated small submodule, M is finitely generated
by Lemma 1.4.
We can to finish the proof of Theorem 2.1. Lemma 2.3 claims that
there exists an infinitely generated small S-torsion-free module which is a
factor of M . As it contradicts to Lemma 2.4, T [X] is right steady.
Note that the premise of Theorem 2.1 that T is finitely generated over
its center is necessary only in Lemma 2.4, in another words, Lemmas 2.2
and 2.3 holds true for a general skew field T .
Recall that an ideal I is right T-nilpotent if for every sequence {ai}i<ω
of elements of I there exists n such that an · . . . · a1 = 0. As every right
T-nilpotent ideal is right steady in the sense of paper [12], we may prove
the following three assertions.
Lemma 2.5. Let R be a ring, X a set of variables and I a right T-
nilpotent ideal of R. Then R[X] is right steady iff (R/J(R))[X] is right
steady.
J. Žemlička 115
Proof. Since (R/J(R))[X] is isomorphic to a factor of R[X], it is enough
to prove the reverse implication. Assume that there exists an infinitely
generated small R[X]-module M . Note that MJ is an R[X]-submodule of
M because all variables commutes with elements from J . Moreover, MJ
is superfluous in M as R-module by [1, Lemma 28.3], hence it superfluous
in M as R[X]-module. Thus M/MJ is an infinitely generated small
R/J [X]-module, which finishes the proof.
Proposition 2.6. Let R a be right perfect ring and X a finite set of
variables. Then R[X] is right steady.
Proof. First note that J(R) is right T-nilpotent by [1, Theorem 28.4] and
R/J(R) is semisimple. Applying Lemma 2.5 we see it is enough to check
steadiness of the ring R/J(R)[X]. As R/J(R)[X] is right noetherian by
Hilbert basis theorem, the conclusion follows from [8, 7o] or [4, Proposition
1.9].
Theorem 2.7. Let X be a countable set of variables and R a right perfect
ring such that EndR(S) is finitely generated as a right module over its
center for every simple module S. Then R[X] is right steady.
Proof. Using Lemma 2.5 as in the proof of Proposition 2.6, it suffices to
show that R/J(R)[X] is right steady. By the Wedderburn-Artin theo-
rem the ring R/J(R) is isomorphic to a finite product of matrix rings
Mki(Ti) over finitely generated skew-fields Ti where Ti is isomorphic to
EndR(Si) for a suitable simple R-module Si. Since Ti[X] is right steady
by Theorem 2.1 and right steadiness is a Morita invariant property by
Proposition 1.6(2), Mki(Ti[X]) ∼= Mki(Ti)[X] is right steady. Finally, any
finite product of right steady rings is right steady as well by Proposi-
tion 1.6(3).
3. Non-steady polynomial rings
Recall that a module M is said to be ω1-reducing if for every countably
generated submodule N of M there exists a finitely generated submodule
F of M such that N ⊆ F . Applying Lemma 1.1 it is easy to see that
every ω1-reducing module is an example of a small module. This notion
gives us a natural construction of an infinitely generated small module as
a union of an uncountable chain of finitely generated modules, which we
use in the following example.
Example 3.1. Let R be an arbitrary ring and consider the monoid N
ω1
as the product of ω1 copies of the monoid (N,+, 0) of natural numbers. For
every α < β ≤ ω1 define eαβ ∈ N
ω1 by the rule eαβ(γ) = 1 for γ ∈ 〈α, β)
116 Steadiness of polynomial rings
and eαβ(γ) = 0 elsewhere. Denote by E the submonoid of Nω1 generated
by the set {eαβ | α < β ≤ ω1}.
Now, consider a monoid ring S = R[E]. Put sα = 1eακ ∈ S for every
α < ω1 where 1 is the unit of the ring R. Since sα = sβ · 1eαβ whenever
α < β, (sαS| α < ω1) forms a strictly increasing chain of right principal
ideals of S. Hence the ideal
⋃
α<ω eαS is ω1-generated and ω1-reducing as
a right S-module, which proves that S is not right steady. The symmetric
argument shows that S is not left steady.
As the cardinality of sets of generators of any infinitely generated
monoid is the same as cardinality of the monoid, we may formulate the
following consequence of the previous construction.
Proposition 3.2. If R be a ring and X an uncountable set of variables,
R[X] is neither right nor left steady.
Proof. Since there exists a surjective map of X onto the monoid E from
Example 3.1, it can be extended to a surjective homomorphism from the
free commutative monoid in free generators X to the monoid E. Moreover,
this homomorphism of monoids can be naturally extended to a surjective
homomorphism of the polynomial ring R[X] to the monoid ring R[E].
Since the homomorphic image R[E] of the ring R[X] is neither right nor left
steady by Example 3.1, the conclusion follows by Proposition 1.6(1).
Obviously, any direct sum of an infinitely generated small module
and non-small module is not small, however it has an infinitely generated
small factor. If we look at the reason of non-steadiness of polynomial
rings in uncountably many variables more carefully, i.e. if we explicitly
construct an infinitely generated small module over such a ring, we obtain
an example of non-small module with infinitely generated small factor,
which is far from containing small module as a direct summand.
Example 3.3. Let X be a set of variables of the cardinality ω1 and
Y ⊂ X such that |Y | = |X \Y | (= ω1). Fix a field F , consider the monoid
E as it is defined in 3.1 and fix a surjective map ϕ : X → E such that
ϕ(Y ) = {eακ| α < κ}. Then ϕ can be extended to a ring homomorphism
F [X] → F [E] as it is described in the proof of Proposition 3.2. Now, the
module U =
∑
y∈Y yF [X] is uniform (i.e. it contains no non-trivial direct
summand) because F [X] is an integral domain. As there exists an strictly
increasing chain of subset Yi ⊂ Yi+1 ⊂ Y such that Y =
⋃
i<ω Yi, the
module U =
⋃
i<ω
∑
y∈Yi
yF [X] is not small by Lemma 1.1. Since ϕ(U)
has the natural structure of an F [X]-module it is an infinitely generated
small F [X]-module.
J. Žemlička 117
Combining Proposition 3.2 and Theorem 2.7 we obtain the final crite-
rion.
Corollary 3.4. Let X be a set of variables and R a right perfect ring
such that EndR(S) is finitely generated as a right module over its center
for every simple module S. Then R[X] is right steady iff X is countable.
References
[1] F.W. Anderson and K.R. Fuller, Rings and Categories of Modules, 2nd edition,
New York 1992, Springer.
[2] H. Bass Algebraic K-theory, New York 1968, Benjamin.
[3] P.Cohn Free Rings and Their Relations, 2nd ed. London 1985, Academic Press.
[4] R.Colpi and C.Menini: On the structure of ∗-modules, J. Algebra 158 (1993),
400–419.
[5] R. Colpi and J. Trlifaj Classes of generalized ∗-modules, Comm. Algebra 22 (1994),
3985–3995.
[6] P.C. Eklof, K.R. Goodearl and J. Trlifaj: Dually slender modules and steady rings,
Forum Math. 9 (1997), 61–74.
[7] J.L. Gómez Pardo, G. Militaru and C. Năstăsescu, When is HOMR(M,−) equal
to HomR(M,−) in the category R-gr?, Comm. Algebra, 22, (1994), 3171-3181.
[8] R. Rentschler, Sur les modules M tels que Hom(M,−) commute avec les sommes
directes, C.R. Acad. Sci. Paris, 268, 1969, 930–933.
[9] J. Trlifaj: Almost ∗-modules need not be finitely generated, Comm. Algebra, 21
(1993), 2453–2462.
[10] J. Trlifaj: Steady rings may contain large sets of orthogonal idempotents. Abelian
groups and modules (Padova, 1994), Math. Appl., 343, Kluwer Acad. Publ., Dor-
drecht, 1995, 467–473.
[11] J. Trlifaj Strong incompactness for some non-perfect rings, Proc. Amer. Math.
Soc. 123 (1995), 21–25.
[12] J. Žemlička and J. Trlifaj, Steady ideals and rings, Rend. Sem. Mat. Univ. Padova,
98 (1997), 161–172.
[13] J. Žemlička Steadiness is tested by a single module, Contemporary Mathematics,
273 (2001), 301–308.
Contact information
J. Žemlička Department of Algebra, Charles University in
Prague, Faculty of Mathematics and Physics
Sokolovská 83, 186 75 Praha 8, Czech Re-
public
E-Mail: zemlicka@karlin.mff.cuni.cz
URL: www.karlin.mff.cuni.cz
Received by the editors: 10.04.2009
and in final form 03.03.2011.
|