Adjoint functors, preradicals and closure operators in module categories
In this article preradicals and closure operators are studied in an adjoint situation, defined by two covariant functors between the module categories R-Mod and S-Mod. The mappings which determine the relationship between the classes of preradicals and the classes of closure operators of these categ...
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irk-123456789-1884932023-03-03T01:27:19Z Adjoint functors, preradicals and closure operators in module categories Kashu, A.I. In this article preradicals and closure operators are studied in an adjoint situation, defined by two covariant functors between the module categories R-Mod and S-Mod. The mappings which determine the relationship between the classes of preradicals and the classes of closure operators of these categories are investigated. The goal of research is to elucidate the concordance (compatibility) of these mappings. For that some combinations of them, consisting of four mappings, are considered and the commutativity of corresponding diagrams (squares) is studied. The obtained results show the connection between considered mappings in adjoint situation. 2019 Article Adjoint functors, preradicals and closure operators in module categories / A.I. Kashu // Algebra and Discrete Mathematics. — 2019. — Vol. 28, № 2. — С. 260–277. — Бібліогр.: 11 назв. — англ. 1726-3255 2010 MSC: 16D90, 16S90, 18A40, 18E40 06A15 http://dspace.nbuv.gov.ua/handle/123456789/188493 en Algebra and Discrete Mathematics Інститут прикладної математики і механіки НАН України |
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In this article preradicals and closure operators are studied in an adjoint situation, defined by two covariant functors between the module categories R-Mod and S-Mod. The mappings which determine the relationship between the classes of preradicals and the classes of closure operators of these categories are investigated. The goal of research is to elucidate the concordance (compatibility) of these mappings. For that some combinations of them, consisting of four mappings, are considered and the commutativity of corresponding diagrams (squares) is studied. The obtained results show the connection between considered mappings in adjoint situation. |
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Kashu, A.I. |
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Kashu, A.I. Adjoint functors, preradicals and closure operators in module categories Algebra and Discrete Mathematics |
| author_facet |
Kashu, A.I. |
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Kashu, A.I. |
| title |
Adjoint functors, preradicals and closure operators in module categories |
| title_short |
Adjoint functors, preradicals and closure operators in module categories |
| title_full |
Adjoint functors, preradicals and closure operators in module categories |
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Adjoint functors, preradicals and closure operators in module categories |
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Adjoint functors, preradicals and closure operators in module categories |
| title_sort |
adjoint functors, preradicals and closure operators in module categories |
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Інститут прикладної математики і механіки НАН України |
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2019 |
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http://dspace.nbuv.gov.ua/handle/123456789/188493 |
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Adjoint functors, preradicals and closure operators in module categories / A.I. Kashu // Algebra and Discrete Mathematics. — 2019. — Vol. 28, № 2. — С. 260–277. — Бібліогр.: 11 назв. — англ. |
| series |
Algebra and Discrete Mathematics |
| work_keys_str_mv |
AT kashuai adjointfunctorspreradicalsandclosureoperatorsinmodulecategories |
| first_indexed |
2025-07-16T10:34:43Z |
| last_indexed |
2025-07-16T10:34:43Z |
| _version_ |
1837799402024992768 |
| fulltext |
“adm-n4” — 2020/1/24 — 13:02 — page 260 — #110
Algebra and Discrete Mathematics RESEARCH ARTICLE
Volume 28 (2019). Number 2, pp. 260–277
c© Journal “Algebra and Discrete Mathematics”
Adjoint functors, preradicals and closure
operators in module categories
A. I. Kashu
Abstract. In this article preradicals and closure opera-
tors are studied in an adjoint situation, defined by two covariant
functors between the module categories R-Mod and S-Mod. The
mappings which determine the relationship between the classes of
preradicals and the classes of closure operators of these categories
are investigated. The goal of research is to elucidate the concordance
(compatibility) of these mappings. For that some combinations of
them, consisting of four mappings, are considered and the commuta-
tivity of corresponding diagrams (squares) is studied. The obtained
results show the connection between considered mappings in adjoint
situation.
1. Preliminary notions and facts
The present work is devoted to the study of preradicals and closure
operators in module categories. The behaviour of these constructions is
investigated in an adjoint situation, i.e. in the case of two adjoint covariant
functors between the module categories. There exists a series of mappings
in this case, which realize the relationship between preradicals and closure
operators of considered categories. The principal attention is given to
investigation of compatibility of these mappings, which is expressed as
commutativity of suitable diagrams.
Firstly we will clarify the situation in what we intend to work. Let RUS
be an arbitrary (R,S)-bimodule and consider the following two covariant
2010 MSC: 16D90, 16S90, 18A40, 18E40 06A15.
Key words and phrases: closure operator, adjoint functors, preradical, category
of modules, natural transformation, lattice of submodules.
“adm-n4” — 2020/1/24 — 13:02 — page 261 — #111
A. I. Kashu 261
functors defined by this bimodule,
R-Mod
H=HomR(U,-) // S-Mod,
T=U⊗
S
-
oo
where R-Mod (S-Mod) is the category of left R-modules (S-modules)
and T is left adjoint to H (notation: T ⊣ H) ([1, 2]). By the definition
this means that for every pair of modules X ∈ R-Mod and Y ∈ S-Mod
there exists a natural isomorphism HomR
(
T (Y ), X
)
∼= HomS
(
Y,H(X)
)
.
This adjoint situation is completely defined by two associated natural
transformations,
Φ : TH → ✶R−Mod, Ψ : ✶S−Mod → HT,
which satisfy the following relations:
H(ΦX) ·ΨH(X) = 1H(X), ΦT (Y ) · T (ΨY ) = 1T (Y ),
where X ∈ R-Mod and Y ∈ S-Mod. We remark that any adjoint situation
with covariant functors between two module categories has this form (up
to an isomorphism).
Recall that a preradical r of R-Mod is a subfunctor of the identical
functor of R-Mod, i.e. r is a function which associates to every module
M ∈ R-Mod a submodule r(M) ⊆ M such that f
(
r(M)
)
⊆ r(M ′) for
every R-morphism f :M →M ′ ([3, 4]). Denote by PR(R) the class of all
preradicals of R-Mod. An order relation in PR(R) is defined as follows:
r 6 s⇔ r(M) ⊆ s(M) for every M ∈ R-Mod.
Remind also that a closure operator of R-Mod is a function C, which
associates to every pair N ⊆ M , where N ∈ L(M), a submodule of M
denoted by CM(N), such that the following conditions are satisfied:
(c1) N ⊆ CM(N) (extension);
(c2) IfN1, N2 ∈ L(M) andN1⊆N2, then CM(N1)⊆CM(N2) (monotony);
(c3) For every R-morphism f : M → M ′ and N ∈ L(M) the relation
f
(
CM(N)
)
⊆ CM′
(
f(N)
)
is true (continuity),
where M ∈ R-Mod and L(M) is the lattice of submodules of M ([5–7]).
Let CO(R) be the class of all closure operators of R-Mod with the order
relation: C 6 D ⇔ CM(N) ⊆ DM(N) for every pair N ⊆M of R-Mod.
The relationship between preradicals and closure operators of R-Mod
is expressed by three mappings ([5–7]), which we denote and define as
follows.
“adm-n4” — 2020/1/24 — 13:02 — page 262 — #112
262 Adjoint functors, preradicals, closure operators
a) ϕR : CO(R) → PR(R). For every closure operator C ∈ CO(R) the
corresponding preradical ϕR(C) = rC is defined by the rule:
rC(M)
def
== CM(0) (1.1)
for every M ∈ R-Mod.
b) ψR
1 : PR(R) → CO(R). For every preradical r ∈ PR(R) and every
pair M ⊆ X of R-Mod we define: ψR
1 (r) = Cr, where
C r
X(M)/M
def
== r(X/M). (1.2)
c) ψR
2 : PR(R) → CO(R). For r ∈ PR(R) and M ⊆ X of R-Mod we
define: ψR
2 (r) = Cr, where
(Cr)X(M)
def
== r(X) +M. (1.3)
Then Cr is the greatest among the closure operators C ∈ CO(R) with
the property ϕR(C) = r, while Cr is the least between such operators. If
we define in CO(R) the equivalence relation
C ∼ D ⇐⇒ rC = rD,
then PR(R) ∼= CO(R)/∼, where to every preradical r ∈ PR(R) corre-
sponds the equivalence class [Cr, C
r], i.e. the interval between Cr and Cr.
2. Mappings between preradicals and closure operators
in adjoint situation
In this section we consider the adjoint situation T ⊣ H indicated
above and recall the definitions of the mappings between the preradicals
of categories R-Mod and S-Mod ([8–10]), as well as the definitions of the
mappings between the closure operators of these categories ([5, 11]).
As above, the bimodule RUS defines the adjoint functors
R-Mod
H=HomR(U,-) // S-Mod
T=U⊗
S
-
oo
with the associated transformations Φ : TH →✶R−Mod and Ψ : ✶S−Mod →
HT . In this situation two mappings can be defined
PR(R)
(−)∗ // PR(S)
(−)∗
oo
between the classes of preradicals of studied categories.
“adm-n4” — 2020/1/24 — 13:02 — page 263 — #113
A. I. Kashu 263
a) The mapping r r∗ from PR(R) to PR(S) is defined as follows.
Let r ∈ PR(R) and Y ∈ S-Mod. Applying T and r, we obtain in R-Mod
the sequence
0 → r
(
T (Y )
) ⊆
−−→ T (Y )
πr
T (Y )
−−−−→
nat
T (Y )
/
r
(
T (Y )
)
→ 0,
where πrT (Y ) is the natural epimorphism. Applying H and using Ψ, we
have in R-Mod the composition of morphisms
Y
ΨY
−−−−→ HT (Y )
H(πr
T (Y )
)
−−−−−−→ H[T (Y )
/
r
(
T (Y )
)
].
Preradical r∗ is defined by the rule
r∗(Y )
def
== Ker [H(πrT (Y )) ·ΨY ]. (2.1)
b) Now we define the inverse mapping s s∗ from PR(S) to PR(R).
Let s ∈ PR(S) and X ∈ R-Mod. By H and s we have in S-Mod the
inclusion isH(X) : s
(
H(X)
) ⊆
−−→ H(X). Applying T and using Φ, we
obtain in R-Mod the morphisms:
T
(
s(H(X))
)
T (is
H(X)
)
−−−−−−→ TH(X)
ΦX
−−−→ X.
Preradical s∗ ∈ PR(R) is defined by the rule
s∗(X)
def
== Im[ΦX · T (isH(X))]. (2.2)
In the works [8–10] a series of properties of these mappings is shown.
In continuation we will define two mappings between the classes of
closure operators of the studied categories in adjoint situation ([5, 11]):
CO(R)
(−)∗ // CO(S).
(−)∗
oo
c) We begin with the mapping C C∗ from CO(R) to CO(S). Let
C ∈ CO(R) and n : N
⊆
−−→ Y be an arbitrary inclusion of S-Mod. Apply
T and consider in R-Mod the decomposition of the morphism T (n) by
the operator C:
T (N)
T (n)
))T (n) // ImT (n)
⊆
jnC // CT (Y )
(
ImT (n)
)
⊆
inC // T (Y ),
“adm-n4” — 2020/1/24 — 13:02 — page 264 — #114
264 Adjoint functors, preradicals, closure operators
where T (n) is the restriction of T (n) to its image, and jnC , i
n
C are the
inclusions. Consider the natural epimorphism
πnC : T (Y )
nat
−−→ T (Y )/CT (Y )
(
ImT (n)
)
.
By H and Ψ we obtain in S-Mod the composition of morphisms
Y
ΨY
−−−→ HT (Y )
H(πn
C)
−−−−−−→ H
[
T (Y )/CT (Y )
(
ImT (n)
)]
.
Operator C∗ is defined by the rule:
C∗
Y (N)
def
== Ker [H(πnC) ·ΨY ]. (2.3)
d) Finally, we show the inverse mapping D D∗ from CO(S) to
CO(R). Let D ∈ CO(S) and m : M
⊆
−−→ X be an inclusion of R-Mod.
Apply H and consider the decomposition of H(m) by D:
H(M)
H(m)
''H(m)// ImH(m)
⊆
jmD // DH(X)
(
ImH(m)
)
⊆
imD // H(X),
where H(m) is the restriction of H(m) and jmD , i
m
D are the inclusions.
Returning in R-Mod by T and using Φ, we obtain the composition of
morphisms,
T [DH(X)
(
ImH(m)
)
]
T (imD )
−−−−−→ TH(X)
ΦX
−−−−→ X.
The operator D∗ is defined by the rule:
D∗
X(M)
def
== Im[ΦX · T (imD )] +M. (2.4)
Totalizing the exposed above definitions of the mappings and consid-
ering them together, we obtain the general situation for the pair T ⊣ H
of adjoint functors,
PR(R)
ψR
1
��
ψR
2
��
(−)∗ //
PR(S)
ψS
1
��
ψS
2
��
(−)∗
oo
CO(R)
ϕR
OO
(−)∗ //
CO(S) .
ϕS
OO
(−)∗
oo
“adm-n4” — 2020/1/24 — 13:02 — page 265 — #115
A. I. Kashu 265
The goal of the following investigations consists in the elucidation of
the relations between these ten mappings, in the search of concordance and
compatibility of them. For that we analyze separately diverse combinations
by four mappings (six cases) and we study the commutativity of respective
squares.
In continuation we will consider three pairs of mappings:
I) (ϕR, ϕS), II) (ψR1 , ψ
S
1 ), III) (ψR2 , ψ
S
2 ).
3. Squares containing the first pair of mappings
We start with two combinations of considered mappings in which
participate ϕR and ϕS.
a) Firstly we analyze the square which consists in the following
mappings:
PR(R)
(−)∗ // PR(S)
CO(R)
(−)∗ //
ϕR
OO
CO(S) .
ϕS
OO
Theorem 3.1. For every closure operator C ∈ CO(R) the relation
r∗C = rC∗
is true (in this sense we can say that the previous square is commutative).
Proof. 1) We begin with the route C
ϕR
rC
(−)∗
r∗C for C ∈ CO(R).
The rule (1.1) shows that rC(X)
def
== CX(0) for every X ∈ R-Mod. The
following step rC
(−)∗
r∗C uses the rule (2.1). Namely, for every Y ∈ S-Mod
we have:
(r∗C) (Y )
def
== Ker [H(πrCT (Y )) ·ΨY ], (3.1)
where πrCT (Y ) : T (Y ) → T (Y )
/
rC
(
T (Y )
)
is the natural epimorphism which
leads to the composition of morphisms
Y
ΨY
−−−→ HT (Y )
H(π
rC
T (Y )
)
−−−−−−−−→ H
[
T (Y )
/
rC
(
T (Y )
)]
.
2) For the same operator C ∈ CO(R) now we follow the way: C
(−)∗
C∗ ϕS
rC∗ . The transition C
(−)∗
C∗ is realized by the rule (2.3), i.e. for
every inclusion n : N
⊆
−−→ Y of S-Mod we have C∗
Y (N) = Ker [H(πnC)·ΨY ],
where πnC : T (Y ) → T (Y )/CT (Y )
(
ImT (n)
)
is the natural epimorphism.
“adm-n4” — 2020/1/24 — 13:02 — page 266 — #116
266 Adjoint functors, preradicals, closure operators
On the following step C∗ ϕS
rC∗ we use the definition (1.1): rC∗(Y )
def
==
C∗
Y (0) for every Y ∈ S-Mod. Now we come back to the construction of C∗
and assume N = 0. Then the situation is simplified, since n = 0, T (n) = 0,
ImT (n) = 0, πnC = π0C : T (Y ) → T (Y )/CT (Y )(0), therefore,
rC∗(Y ) = C∗
Y (0)
def
== Ker [H(π0C) ·ΨY ]. (3.2)
3) Now we compare the expressions (3.1) and (3.2) for r∗C(Y ) and
rC∗(Y ). It is obvious that the relation rC
(
T (Y )
) def
== CT (Y )(0) implies the
coincidence of epimorphisms πrCT (Y ) and π0C , so by (3.1) and (3.2) we have
r∗C(Y ) = rC∗(Y ) for every Y ∈ S-Mod, i.e. r∗C = rC∗ .
b) Further we consider the second combination of mappings which
contains ϕR and ϕS, namely the square
PR(R) PR(S)
(−)∗oo
CO(R)
ϕR
OO
CO(S),
(−)∗oo
ϕS
OO
analyzing the concordance of these mappings.
Theorem 3.2. For every closure operator D ∈ CO(S) the relation
rD∗ = r∗D
is true, i.e. the previous square is commutative.
Proof. 1) We begin with the route D
(−)∗
D∗ ϕR
rD∗ , where D ∈ CO(S).
The transition D
(−)∗
D∗ is defined by the rule (2.4), i.e. for every inclusion
m :M
⊆
−−→ X of R-Mod we have: D∗
X(M) = Im[ΦX · T (imD )] +M , where
imD : DH(X)
(
ImH(m)
) ⊆
−−→ H(X) is the inclusion, which leads to the
composition
T [DH(X)
(
ImH(m)
)
]
T (imD )
−−−−−→ TH(X)
ΦX
−−−−→ X.
The following step D∗ ϕR
rD∗ is defined by the rule (1.1), i.e. rD∗(X)
def
==
D∗
X(0) for everyX ∈ R-Mod. To specify the module D∗
X(0) we assume M =
0 in the above construction of D∗
X(M). Then m = 0,H(m) = 0,H(m) = 0,
TH(M) = 0, T
(
ImH(m)
)
= 0, therefore DH(X)
(
ImH(m)
)
= DH(X)(0),
“adm-n4” — 2020/1/24 — 13:02 — page 267 — #117
A. I. Kashu 267
imD = i0D : DH(X)(0)
⊆
−−→ H(X) and T (i0D) : T
(
DH(X)(0)
)
→ TH(X). In
such a way we obtain
rD∗(X) = D∗
X(0) = Im[ΦX · T (i0D)]. (3.3)
2) Further we follow the way D
ϕS
rD
(−)∗
r∗D for D ∈ CO(S). By the
rule (1.1) we have rD(Y )
def
== DY (0) for every Y ∈ S-Mod. The second
step rD
(−)∗
r∗D is defined by the rule (2.2), i.e. for every X ∈ R-Mod we
have
r∗D(X) = Im[ΦX · T (irDH(X))], (3.4)
where the inclusion irDH(X) : rD
(
H(X)
) ⊆
−−→ H(X) implies the composition
T
[
rD
(
H(X)
)]
T (i
rD
H(X)
)
−−−−−−−→ TH(X)
ΦX
−−−−→ X.
3) Now we compare the expressions (3.3) and (3.4). Since by the
definition rD
(
H(X)
) def
== DH(X)(0), it is clear that the inclusions i0D and
irDH(X) coincide, so by the indicated above relations it follows that rD∗(X) =
r∗D(X) for every X ∈ R-Mod, i.e. rD∗ = r∗D.
4. Squares containing the second pair of mappings
In continuation we analyze two combinations of the studied mappings
in which participate ψR1 and ψS1 .
a) Now we consider the square
PR(R)
ψR
1
��
(−)∗ // PR(S)
ψS
1
��
CO(R)
(−)∗ // CO(S) .
Theorem 4.1. For every preradical r ∈ PR(R) the relation
Cr
∗
= (Cr)∗
is true, i.e. the previous diagram is commutative.
Proof. 1) Let r ∈ PR(R). Firstly we follow the route: r
(−)∗
r∗
ψS
1
Cr
∗
.
The translation r
(−)∗
r∗ is defined by (2.1), i.e. for every Y ∈ S-Mod we
“adm-n4” — 2020/1/24 — 13:02 — page 268 — #118
268 Adjoint functors, preradicals, closure operators
have: r∗(Y )
def
== Ker [H(πrT (Y )) ·ΨY ], where πrT (Y ) : T (Y ) → T (Y )
/
r
(
T (Y )
)
is the natural epimorphism, which defines the composition
Y
ΨY
−−−→ HT (Y )
H(πr
T (Y )
)
−−−−−−−→ H
[
T (Y )
/
r
(
T (Y )
)]
.
The following step r∗
ψS
1
C r∗ is defined by (1.2), i.e. for every inclusion
n : N
⊆
−−→ Y of S-Mod we have: Cr
∗
Y (N)
/
N
def
== r∗(Y/N). To precise the
expression of r∗(Y/N), in the above construction of r∗ we substitute Y
by Y/N . Then we obtain the natural epimorphism
πrT (Y/N) : T (Y/N)
nat
−−→ T (Y/N)
/
r
(
T (Y/N)
)
and the composition
Y/N
ΨY/N
−−−−→ HT (Y/N)
H
(
πr
T (Y/N)
)
−−−−−−−−−−→ H
[
T (Y/N)
/
r
(
T (Y/N)
)]
.
By the definition we have r∗(Y/N) = Ker [H(πrT (Y/N)) · ΨY/N ], therefore
Cr
∗
Y = Ker [H(πrT (Y/N)) · ΨY/N ]. Denoting by πN : Y → Y/N the natural
epimorphism, now it is easy to see that
Cr
∗
Y (N) = Ker [H(πrT (Y/N)) ·ΨY/N · πN ]. (4.1)
2) Further, for r ∈ PR(R) we consider the way: r
ψR
1
Cr
(−)∗
(Cr)∗.
The first step r
ψR
1
Cr is defined by (1.2), i.e. for every pair M ⊆ X of
R-Mod we have C r
X(M)
/
M
def
== r(X/M). The transition Cr
(−)∗
(Cr)∗ is
determined by (2.3). This means that for every inclusion n : N
⊆
−−→ Y of S-
Mod we consider in R-Mod the decomposition of T (n) by the operator Cr:
T (N)
T (n)
**T (n) // ImT (n)
⊆
jn
C
r // Cr
T (Y )
(
ImT (n)
)
⊆
in
C
r // T (Y ).
By the natural epimorphism πnCr : T (Y )
nat
−−→ T (Y )
/
CrT (Y )
(
ImT (n)
)
we
obtain in S-Mod the composition
Y
ΨY
−−−→ HT (Y )
H(πn
Cr )
−−−−−−−→ H
[
T (Y )
/
CrT (Y )
(
ImT (n)
)]
.
Using (2.3) we have
(Cr)∗Y (N)
def
== Ker [H(πnCr) · ψY ]. (4.2)
“adm-n4” — 2020/1/24 — 13:02 — page 269 — #119
A. I. Kashu 269
3) Now we verify the relation between Cr
∗
Y (N) and (Cr)∗Y (N). For
that we consider in S-Mod the diagram
Cr
∗
Y (N)
π′
N
nat
//
⊇
��
r∗(Y/N) = Cr
∗
Y (N)/N
⊇
��
N
n
⊆
//
ΨN
��
Y
ΨY
��
��
πN
nat
// Y/N
��
ΨY/N
��
HT (N)
HT (n) // HT (Y )
H(πn
Cr )
��
HT (πN ) // HT (Y/N)
H(πr
T (Y/N)
)
��
H[T (Y )
/
CrT (Y )
(
ImT (n)
)
]
∼= // H[T (Y/N)
/
r
(
T (Y/N)
)
],
where π′N is the natural epimorphism and r∗(Y/N)
def
== Cr
∗
Y (N)/N. We
search the relation between the modules of the last line. For that we
look for the connection between the modules T (Y )
/
CrT (Y )
(
ImT (n)
)
and
T (Y/N)
/
r
(
T (Y/N)
)
. Since T is right exact, it transforms the short ex-
act sequence 0 −→ N
n
−−→
⊆
Y
πN
−−→
nat
Y/N → 0 in an exact sequence
T (N)
T (n)
−−−→ T (Y )
T (πN )
−−−−→ T (Y/N) → 0, therefore we have the exact se-
quence 0 → ImT (n)
⊆
−→ T (Y )
T (πN )
−−−−→ T (Y/N) → 0. Then it is clear that
T (Y/N) ∼= T (Y )/ ImT (n), which implies the isomorphism
T (Y/N)
/
r
(
T (Y/N)
)
∼= [T (Y )/ ImT (n)]
/
r[T (Y )/ ImT (n)].
Using (1.2) for Cr, we have r[T (Y )/ ImT (n)]=CrT (Y )
(
ImT (n)
)
/ ImT (n),
and substituting this module in the previous relation we obtain
T (Y/N)
/
r
(
T (Y/N)
)
∼= [T (Y )/ ImT (n)]
/
[CrT (Y )
(
ImT (n)
)/
ImT (n)]
∼= T (Y )
/
CrT (Y )
(
ImT (n)
)
.
Applying H now we have in S-Mod the isomorphism
H[T (Y )
/
CrT (Y )
(
ImT (n)
)
] ∼= H[T (Y/N)
/
r
(
T (Y/N)
)
],
which closes the previous diagram. Therefore,
Ker [H(πnCr) ·ΨY ] = Ker [H(πrT (Y/N)) ·ΨY/N · πN ]
and by (4.1) and (4.2) this means that (Cr)∗Y (N) = Cr
∗
Y (N) for every
inclusion N ⊆ Y of S-Mod. Thus (Cr)∗ = Cr
∗
.
“adm-n4” — 2020/1/24 — 13:02 — page 270 — #120
270 Adjoint functors, preradicals, closure operators
b) In continuation we consider the square
PR(R)
ψR
1
��
PR(S)
ψS
1
��
(−)∗oo
CO(R) CO(S)
(−)∗oo
and verify the concordance of his mappings.
Theorem 4.2. For every preradical s ∈ PR(S) the relation
(Cs)∗ 6 Cs
∗
is true. If the module RU is projective, then (Cs)∗ = Cs
∗
, i.e. the studied
square is commutative.
Proof. 1) Let s ∈ PR(S). We begin with the route: s
(−)∗
s∗
ψR
1
Cs
∗
.
The transition s
(−)∗
s∗ is defined by (2.2), i.e. for every X ∈ R-Mod
we have the inclusion isH(X) : s
(
H(X)
) ⊆
−−→ H(X), which leads to the
composition T
[
s
(
H(X)
)]
T (is
H(X)
)
−−−−−−→ TH(X)
ΦX
−−→ X. By the rule (2.2) we
have s∗(X)
def
== Im[ΦX · T (isH(X))].
The following step s∗
ψR
1
Cs
∗
is realized by the rule (1.2), i.e. for every
inclusion M ⊆ X of R-Mod we have: Cs
∗
X (M)
/
M = s∗(X/M). In the
above construction of s∗ we substitute the module X by X/M , obtaining
the inclusion isH(X/M) : s
(
H(X/M)
) ⊆
−−→ H(X/M) and the composition
T
[
s
(
H(X/M)
)]
T (is
H(X/M)
)
−−−−−−−→ TH(X/M)
ΦX/M
−−−→ X/M.
By the definition (1.2) in this case we have
s∗(X/M)
def
== Im[ΦX/M · T (isH(X/M))],
therefore,
C
s∗
X (M)
/
M = Im[ΦX/M · T (isH(X/M))]. (4.3)
2) For s ∈ PR(S) now we consider the transitions: s
ψS
1
Cs
(−)∗
(Cs)∗.
The operator Cs is obtained by (1.2), i.e. CsY (N)
/
N
def
== s(Y/N) for every
inclusion N ⊆ Y of S-Mod.
“adm-n4” — 2020/1/24 — 13:02 — page 271 — #121
A. I. Kashu 271
Further, for the step Cs
(−)∗
(Cs)∗ we use the rule (2.4). Namely,
for every inclusion m : M
⊆
−−→ X of R-Mod we consider the inclu-
sion imCs : CsH(X)
(
ImH(m)
) ⊆
−−→ H(X) of S-Mod and the composition
T
[
CsH(X)
(
ImH(m)
)] T (im
Cs )
−−−−→ TH(X)
ΦX
−−→ X in R-Mod.
By (2.4) we have
(Cs)∗X(M)
def
== Im[ΦX · T (imCs)] +M. (4.4)
We mention also that by the definition of Cs for the pair ImH(m) ⊆ H(X)
we have
CsH(X)
(
ImH(m)
)/
ImH(m)
def
== s
(
H(X)/ ImH(m)
)
.
3) It remains to compare the obtained expressions for C
s∗
X (M) and
(Cs)∗X(M). To this end we consider in S-Mod the commutative diagram
H(M)
H(m)
��
H(m) // H(X)
H(πX) // H(X/M)
ImH(m)
jmCs⊇
��
⊆ // H(X)
π
nat
// H(X)/ ImH(m)
∼=
−−−−→ ImH(πX)
⊆ // H(X/M)
CsH(X)
(
ImH(m)
)
⊆
im
Cs
BB
f
22
π′
nat
// A
⊆
@@
def
s
(
H(X)/ ImH(m)
)
⊆
QQ
ϕ′
// s
(
H(X/M)
)
,
⊆
OO
where A = CsH(X)
(
ImH(m)
)
/ ImH(m), the first line is the image of
the short exact sequence 0 //M
m
⊆
// X
πX
nat
// X/M // 0, and π, π′
are natural epimorphisms. Denoting ϕ : H(X)/ ImH(m)
∼=
−−−→ ImH(πX)
⊆
−−→ H(X/M), we have its restriction ϕ′ (by definition of preradical) and
f = ϕ′ · π′.
“adm-n4” — 2020/1/24 — 13:02 — page 272 — #122
272 Adjoint functors, preradicals, closure operators
Applying T to this diagram and using Φ, we obtain in R-Mod the
following commutative diagram
(Cs)∗X(M)
⊆ //
⊇
��
Cs
∗
X (M)
π′
X
nat
// s∗(X/M) = C
s∗
X (M)/M
⊇
��
X
πX
nat
// X/M
TH(X)
ΦX
OO
TH(πX)
// TH(X/M)
ΦX/M
OO
T [CsH(X)
(
ImH(m)
)
]−
T (im
Cs )
OO
BB
T (f) // T [s
(
H(X/M)
)
] .
T
(
is
H(X/M)
)
OO
@@
By the mentioned above relations we have (Cs)∗X(M) = Im[ΦX ·T (i
m
Cs)]+M
and s∗(X/M)
def
== Cs
∗
X (M)/M = Im[ΦX/M · T (isH(X/M))]. The commutativ-
ity of this diagram implies that
πX · ΦX · T (imCs) = ΦX/M · T (isH(X/M)) · T (f).
Therefore,
Im[πX · ΦX · T (imCs)] = Im[ΦX/M · T (isH(X/M)) · T (f)]
⊆ Im[ΦX/M · T (isH(X/M))]
def
== s∗(X/M) = Cs
∗
X (M)/M.
Since Im[πX · ΦX · T (imCs)] = πX
(
Im[ΦX · T (imCs)]
)
= (Im[ΦX · T (imCs) +
M ])/M
def
== [(Cs)∗X(M)]/M , from the previous relation it follows
that [(Cs)∗X(M)]/M ⊆ Cs
∗
X (M)/M . Therefore (Cs)∗X(M) ⊆ Cs
∗
X (M) for
every M ⊆ X, which means that (Cs)∗ 6 Cs
∗
, proving the first statement
of the theorem.
4) Now we will prove the second statement, assuming that RU is
a projective module. Since H = HomR(U, -), this means that H is an
exact functor, i.e. every epimorphism π of R-Mod is transformed into an
epimorphism H(π) of S-Mod.
Following the above proof, we observe that since πX is an epimorphism,
in this case H(πX) is an epimorphism, i.e. ImH(πX) = H(X/M). Then ϕ
is an isomorphism, therefore ϕ′ is an isomorphism. But then f = ϕ′ · π′ is
an epimorphism, hence T (f) is an epimorphism (the functor T is right
exact). Therefore from the last diagram is clear that
Im[ΦX/M · T (isH(X/M)) · T (f)] = Im[ΦX/M · T (isH(X/M))].
“adm-n4” — 2020/1/24 — 13:02 — page 273 — #123
A. I. Kashu 273
From the proof of the first part now it is obvious that in this case
instead of inclusion we obtain the equality (Cs)∗X(M) = Cs
∗
X (M), so
(Cs)∗ = Cs
∗
.
5. Squares containing the third pair of mappings
In this section we consider the last two cases and we study the combi-
nations of the mappings which contain ψR
2 and ψS
2 .
a) We will examine the following square:
PR(R)
ψR
2
��
(−)∗ // PR(S)
ψS
2
��
CO(R)
(−)∗ // CO(S) ,
verifying the compatibility of his mappings.
Theorem 5.1. For every preradical r ∈ PR(R) the relation
Cr∗ 6 C
∗
r
is true.
Proof. 1) Let r ∈ PR(R) and consider the way: r
(−)∗
r∗
ψS
2
Cr∗ . For
Y ∈ S-Mod applying T and r we obtain in R-Mod the sequence
0 // r
(
T (Y )
) ⊆ // T (Y )
πr
T (Y )
nat
// T (Y )
/
r
(
T (Y )
)
// 0.
Using H and Ψ, we have in S-Mod the composition
Y
ΨY // HT (Y )
H(πr
T (Y )
)
// H[T (Y )
/
r
(
T (Y )
)
],
and by the rule (2.1) we obtain: r∗(Y )
def
== Ker [H(πrT (Y )) ·ΨY ].
The following step r∗
ψS
2
Cr∗ is defined by (1.3), i.e. for every inclusion
n : N
⊆
−−→ Y of S-Mod we have
(Cr∗)Y (N)
def
== r∗(Y ) +N = Ker [H(πrT (Y )) ·ΨY ] +N. (5.1)
2) For r ∈ PR(R) now we follow the route: r
ψR
2
Cr
(−)∗
C∗
r . The
operator Cr is defined by the rule (1.3): (Cr)X(M)
def
== r(X)+M for every
“adm-n4” — 2020/1/24 — 13:02 — page 274 — #124
274 Adjoint functors, preradicals, closure operators
inclusion M ⊆ X of R-Mod. Further, the transition Cr
(−)∗
C∗
r is defined
by (2.3). Namely, for an inclusion n : N
⊆
−−→ Y of S-Mod, applying T
and Cr, we obtain in R-Mod the situation
0→(Cr)T (Y )(ImT (n))
inCr
−−−→
⊆
T (Y )
πn
Cr
−−−→
nat
T (Y )
/
(Cr)T (Y )(ImT (n)) → 0,
and by (1.3) we have (Cr)T (Y )
(
ImT (n)
) def
== r
(
T (Y )
)
+ ImT (n).
Returning back in S-Mod by H, we obtain the composition
Y
ΨY
−−−−→ HT (Y )
H(πn
Cr
)
−−−−−−→ H[T (Y )
/
(Cr)T (Y )
(
ImT (n)
)
].
By the rule (2.3) we have
(C∗
r )Y (N)
def
== Ker [H(πnCr
) ·ΨY ]. (5.2)
3) To compare the modules (Cr∗)Y (N) and (C∗
r )Y (N), indicated in
(5.1) and (5.2), we consider in R-Mod the situation
r
(
T (Y )
)
i
⊆
''
j⊇
��
T (Y )
/
r
(
T (Y )
)
π
��
T (N)
T (n) // T (Y )
π
r
T (Y
)
na
t
55
πn
Cr
nat
))
r
(
T (Y )
)
+ ImT (n)
i
n
Cr
⊆
77
T (Y )
/(
r(T (Y )) + ImT (n)
)
,
where r(T (Y )) + ImT (n) = (Cr)T (Y )
(
ImT (n)
)
. The inclusion j :
r
(
T (Y )
) ⊆
−−→ r
(
T (Y )
)
+ ImT (n) implies the epimorphism π such that
the above diagram is commutative. Applying H we obtain in R-Mod the
commutative diagram
H
(
r
(
T (Y )
))
H(i)
**
H(j)
��
H[T (Y )
/
r
(
T (Y )
)
]
H(π)
��
Y
ΨY // H(T (Y ))
H(π
r
T (Y
)
) 44
H(πn
Cr
)
**
H[r
(
T (Y )
)
+ ImT (n)]
H(i
n
Cr
) 44
H[T (Y )
/(
r(T (Y )) + ImT (n)
)
].
The commutativity of the right triangle implies Ker [H(π) ·H(πrT (Y ))] =
KerH(πnCr
). Therefore KerH(πrT (Y )) ⊆ KerH(πnCr
), so the relation
“adm-n4” — 2020/1/24 — 13:02 — page 275 — #125
A. I. Kashu 275
Ker [H(πrT (Y )) · ΨY ] ⊆ Ker [H(πnCr
) · ΨY ]
def
== (C∗
r )Y (N) is true. Since
N ⊆ (C∗
r )Y (N), it is obvious that Ker [H(πrT (Y )) · ΨY +N ] ⊆ (C∗
r )Y (N).
Now by (5.1) we have (Cr∗)Y (N) ⊆ (C∗
r )Y (N) for every N ⊆ Y , i.e.
Cr∗ 6 C
∗
r .
b) The last possible square consisting of the studied mappings is the
following:
PR(R)
ψR
2
��
PR(S)
ψS
2
��
(−)∗oo
CO(R) CO(S) .
(−)∗oo
As usual, we examine the relation between these mappings.
Theorem 5.2. For every preradical s ∈ PR(S) the relation
Cs∗ 6 C
∗
s
is true.
Proof. 1) Let s ∈ PR(S) and consider the way: s
ψS
2
Cs
(−)∗
C∗
s . By the
rule (1.3) we have the operator Cs such that (Cs)Y (N)
def
== s(Y ) +N for
every inclusion N ⊆ Y of S-Mod.
Further, the transition Cs
(−)∗
C∗
s from CO(S) to CO(R) is defined
by the rule (2.4). This means that for every inclusion m :M
⊆
−−→ X of
R-Mod we consider the decomposition of H(m) by the operator Cs,
H(M)
H(m)
''H(m)// ImH(m)
⊆
jmCs // (Cs)H(X)
(
ImH(m)
)
⊆
imCs // H(X),
where (Cs)H(X)
(
ImH(m)
) def
== s
(
H(X)
)
+ ImH(m). Applying T , we
obtain in R-Mod the composition of morphisms
T [s
(
H(X)
)
+ ImH(m)] = T [(Cs)H(X)(ImH(m))]
T (imCs
)
−−−−→ TH(X)
ΦX
−−→X.
By the definition (2.4) we have
(C∗
s )X(M)
def
== Im[ΦX · T (imCs
)] +M. (5.3)
“adm-n4” — 2020/1/24 — 13:02 — page 276 — #126
276 Adjoint functors, preradicals, closure operators
2) For the same preradical s ∈ PR(S) now we consider the transitions
s
(−)∗
s∗
ψR
2
Cs∗ . To obtain s∗, let X ∈ R-Mod for which we have the
inclusion isH(X) : s
(
H(X)
) ⊆
−−−→ H(X). Applying T and using Φ, we
obtain in R-Mod the composition of morphisms
T [s
(
H(X)
)
]
T (is
H(X)
)
−−−−−−→ TH(X)
ΦX
−−→ X.
The preradical s∗ is defined by the rule (2.2), i.e.,
s∗(X)
def
== Im[ΦX · T (isH(X))].
For the following transition s∗
ψR
2
Cs∗ the rule (1.3) is used: (Cs∗)X(M)
def
== s∗(X) +M for every pair M ⊆ X of R-Mod. Taking into account the
form of s∗(X) indicated above, now we have
(Cs∗)X(M) = Im[ΦX · T (isH(X))] +M. (5.4)
3) Finally, we compare the constructions of parts 1) and 2). In S-Mod
we have the inclusions
s
(
H(X)
)
i s
H(X)
⊆
++
i⊇
��
H(X),
s
(
H(X)
)
+ ImH(m)
i
m
Cs
⊆
33
which implies in R-Mod the situation
T [s
(
H(X)
)
] T (i s
H(X) )
((
T (i)
��
(Cs∗)X(M)
def
== Im[ΦX · T (isH(X))]+M
⊇
uu
⊇
��
TH(X)
ΦX // X
T [s
(
H(X)
)
+ ImH(m)]
T (i
m
Cs
) 66
(C∗
s )X(M)
def
== Im[ΦX · T (imCs
)]+M.
⊇
ii
The commutativity of the left triangle implies ImT (isH(X)) ⊆ ImT (imCs
),
therefore Im[ΦX · T (isH(X))] ⊆ Im[ΦX · T (imCs
)]. By (5.3) and (5.4)
this means that (Cs∗)X(M) ⊆ (C∗
s )X(M) for every M ⊆ X, i.e.
Cs∗ 6 C
∗
s .
“adm-n4” — 2020/1/24 — 13:02 — page 277 — #127
A. I. Kashu 277
In conclusion we can affirm that the indicated ten mappings, which
realize the connection between preradicals and closure operators in adjoint
situation, are well concordant between them. For the six combinations,
which constitute the squares of mappings, in three cases the commutativity
of respective diagrams is proved (Theorems 3.1, 3.2, 4.1), and in other
three cases the inclusion relations are shown (Theorems 4.2, 5.1, 5.2).
References
[1] F. W. Anderson, K. R. Fuller, Rings and categories of modules, Springer - Verlag,
New York, 1992.
[2] R. Wisbauer, Foundations of module and ring theory, Gordon and Breach Science
Publishers, 1991.
[3] L. Bican, T. Kepka, P. Nemec, Rings, modules and preradicals, Marcel Dekker,
New York, 1982.
[4] P. Gabriel, Des categories abeliennes, Bull. Soc. Math. France, 90, 1962,
pp. 323–448.
[5] D. Dikranjan, V. Tholen, Categorical structure of clsure operators, Kluwer Academic
Publishers, 1995.
[6] D. Dikranjan, E. Giuli, Factorizations, injectivity and completness in categories of
modules, Commun. in Algebra, 19, N.1, 1991, pp. 45–83.
[7] D. Dikranjan, E. Giuli, Closure operators I, Topology and its Applications, 27,
1987, pp. 129–143.
[8] A. I. Kashu, Preradicals in adjoint situation, Matem. issled. (Kishinev), 48, 1978,
pp. 48-64 (In Russian).
[9] A. I. Kashu, On correspondence of preradicals and torsions in adjoint situation,
Matem. issled. (Kishinev), 56, 1980, pp. 62-84 (In Russian).
[10] A. I. Kashu, Functors and torsions in categories of modules, Academy of Sciences
of Moldova, Institute of Mathematics, Khishinev, 1997 (In Russian).
[11] A. I. Kashu, Closure operators in modules and adjoint functors I, Algebra and
Discrete Mathematics, 25, N.1, 2018, pp. 98–117.
Contact information
A. I. Kashu Institute of Mathematics and Computer,
Science ”Vladimir Andrunachievici”, Academiei
str., 5, MD–2028, Kishinev, Moldova
E-Mail(s): alexei.kashu@math.md
Received by the editors: 21.01.2019.
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