$\scr{Z^{ \ast}}$ - semilocal modules and the proper class $\scr{RS}$
Over an arbitrary ring, a module $M$ is said to be $\scr{Z^{ \ast}}$ -semilocal if every submodule $U$ of $M$ has a $\scr{Z^{ \ast}}$ -supplement $V$ in $M$, i.e., $M = U + V$ and $U \cap V \subseteq \scr{Z^{ \ast}} (V )$, where $\scr{Z^{ \ast}}(V ) = \{m \in V | Rm$ is a small module $\}$ is...
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860507201349615616 |
|---|---|
| author | Türkmen, E. Тюркмен, Є. |
| author_facet | Türkmen, E. Тюркмен, Є. |
| author_sort | Türkmen, E. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2019-12-05T08:55:13Z |
| description | Over an arbitrary ring, a module $M$ is said to be $\scr{Z^{ \ast}}$ -semilocal if every submodule $U$ of $M$ has a $\scr{Z^{ \ast}}$ -supplement $V$
in $M$, i.e., $M = U + V$ and $U \cap V \subseteq \scr{Z^{ \ast}} (V )$, where $\scr{Z^{ \ast}}(V ) = \{m \in V | Rm$ is a small module $\}$ is the $\mathrm{R}\mathrm{a}\mathrm{d}$-small
submodule. In this paper, we study basic properties of these modules as a proper generalization of semilocal modules. In particular, we show that the class of $\scr{Z^{ \ast}}$ -semilocal modules is closed under submodules, direct sums, and factor modules.
Moreover, we prove that a ring $R$ is $\scr{Z^{ \ast}}$ -semilocal if and only if every injective left R-module is semilocal. In addition,
we show that the class $\scr{RS}$ of all short exact sequences $E :0 \xrightarrow{\psi} M \xrightarrow{\phi} K \rightarrow 0$ such that $\mathrm{I}\mathrm{m}(\psi )$ has a
$\scr{Z^{ \ast}}$ -supplement in $N$ is a proper class over left hereditary rings. We also study some homological objects of the proper
class $\scr{RS}$ . |
| first_indexed | 2026-03-24T02:05:33Z |
| format | Article |
| fulltext |
UDC 512.5
E. Türkmen (Amasya Univ., Turkey)
\bfscrZ \ast -SEMILOCAL MODULES AND THE PROPER CLASS \bfscrR \bfscrS
\bfscrZ \ast -НАПIВЛОКАЛЬНI МОДУЛI ТА ВЛАСНИЙ КЛАС \bfscrR \bfscrS
Over an arbitrary ring, a module M is said to be \scrZ \ast -semilocal if every submodule U of M has a \scrZ \ast -supplement V
in M, i.e., M = U + V and U \cap V \subseteq \scrZ \ast (V ), where \scrZ \ast (V ) = \{ m \in V | Rm is a small module\} is the \mathrm{R}\mathrm{a}\mathrm{d}-small
submodule. In this paper, we study basic properties of these modules as a proper generalization of semilocal modules. In
particular, we show that the class of \scrZ \ast -semilocal modules is closed under submodules, direct sums, and factor modules.
Moreover, we prove that a ring R is \scrZ \ast -semilocal if and only if every injective left R-module is semilocal. In addition,
we show that the class \scrR \scrS of all short exact sequences \BbbE : 0 - \rightarrow M
\psi - \rightarrow N
\phi - \rightarrow K - \rightarrow 0 such that \mathrm{I}\mathrm{m}(\psi ) has a
\scrZ \ast -supplement in N is a proper class over left hereditary rings. We also study some homological objects of the proper
class \scrR \scrS .
Над довiльним кiльцем модуль M називається \scrZ \ast -напiвлокальним, якщо кожний пiдмодуль U модуля M має
\scrZ \ast -доповнення V в M, тобто M = U + V i U \cap V \subseteq \scrZ \ast (V ), де \scrZ \ast (V ) = \{ m \in V | Rm — малий модуль\} —
\mathrm{R}\mathrm{a}\mathrm{d}-малий пiдмодуль. У цiй роботi вивчаються базовi властивостi таких модулiв, як вiдповiдного узагальнення
напiвлокальних модулiв. Зокрема, показано, що клас \scrZ \ast -напiвлокальних модулiв є замкненим вiдносно пiдмодулiв,
прямих сум i фактор-модулiв. Крiм того, доведено, що кiльце R є \scrZ \ast -напiвлокальним тодi i тiльки тодi, коли кожен
iн’єктивний лiвий R-модуль є напiвлокальним. Також встановлено, що клас \scrR \scrS усiх коротких послiдовностей \BbbE :
0 - \rightarrow M
\psi - \rightarrow N
\phi - \rightarrow K - \rightarrow 0 таких, що \mathrm{I}\mathrm{m}(\psi ) має \scrZ \ast -доповнення в N, є власним класом над лiвими спадковими
кiльцями. Вивчено також деякi гомологiчнi об’єкти власного класу \scrR \scrS .
1. Introduction. Throughout this study, all rings are associative with identity and all modules are
unital left R-modules. Let R be a ring and M be a left R-module. The Jacobson radical of M will
be denoted by \mathrm{R}\mathrm{a}\mathrm{d}(M), and the injective hull of the module M will be denoted by E(M). The
notation N \subseteq M (N \subset M ) means that N is a (proper) submodule of M. A non-zero submodule
L \subseteq M is said to be essential in M, denoted as L\unlhd M, if L\cap N \not = 0 for every non-zero submodule
N \subseteq M. Dually, a proper submodule N \subset M is said to be small in M, denoted by N \ll M, if
M \not = N + K for every proper submodule K of M (see [14], 19.1). A module M is said to be
small if M is a small submodule of some R-module (see [7]). It is shown in [7] (Theorem 1) that
a module M is small if and only if M is a small submodule of E(M). It is clear that every small
submodule of M is a small module. For a module M, we consider the following submodule of M :
\scrZ \ast (M) = \{ m \in M | Rm is a small module\} .
Since \mathrm{R}\mathrm{a}\mathrm{d}(M) is the sum of all small submodules of M, we get \mathrm{R}\mathrm{a}\mathrm{d}(M) \subseteq \scrZ \ast (M). It is easy to see
that \scrZ \ast (M) =M\cap \mathrm{R}\mathrm{a}\mathrm{d}(E(M)). Clearly, \scrZ \ast (M) =M if and only if M \subseteq \mathrm{R}\mathrm{a}\mathrm{d}(E(M)). A module
M is said to be \mathrm{R}\mathrm{a}\mathrm{d}-small (according to [13], cosingular) if \scrZ \ast (M) = M. Since \scrZ \ast (\scrZ \ast (M)) =
= \scrZ \ast (M), \scrZ \ast (M) is the largest \mathrm{R}\mathrm{a}\mathrm{d}-small submodule of M. Small modules are \mathrm{R}\mathrm{a}\mathrm{d}-small. Also,
a finitely generated \mathrm{R}\mathrm{a}\mathrm{d}-small module is small.
Let M be a module and U, V \subseteq M be submodules. V is called a supplement (\mathrm{R}\mathrm{a}\mathrm{d}-supplement,
respectively) of U in M if M = U + V and U \cap V \ll V (U \cap V \subseteq \mathrm{R}\mathrm{a}\mathrm{d}(V )). M is called
supplemented (\mathrm{R}\mathrm{a}\mathrm{d}-supplemented, respectively) if every submodule of M has a (\mathrm{R}\mathrm{a}\mathrm{d}-) supplement
in M. Characterizations and structures of supplemented and \mathrm{R}\mathrm{a}\mathrm{d}-supplemented modules are ex-
c\bigcirc E. TÜRKMEN, 2019
400 ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 3
\scrZ \ast -SEMILOCAL MODULES AND THE PROPER CLASS \scrR \scrS 401
tensively studied by many authors. We specifically mention [4, 14, 15] among papers concerning
supplemented and \mathrm{R}\mathrm{a}\mathrm{d}-supplemented modules.
Since \mathrm{R}\mathrm{a}\mathrm{d}(V ) \subseteq \scrZ \ast (V ), it is natural to introduce another notion that we called a submodule
V of M a \scrZ \ast -supplement of U in M provided M = U + V and U \cap V \subseteq \scrZ \ast (V ). Follo-
wing [13] (Lemma 2.6 and Proposition 3.10), we characterize modules whose submodules have a
\scrZ \ast -supplement.
Lemma 1.1. Let R be a ring and M be an R-module. Then the following statements are
equivalent:
(1) Every submodule U of M has a \scrZ \ast -supplement V in M.
(2) For any submodule U of M, there exists a submodule V of M such that M = U + V and
U \cap V \subseteq \scrZ \ast (M).
(3) If U is a submodule of M, then M = U + V and U \cap V is \mathrm{R}\mathrm{a}\mathrm{d}-small for some submodule
V of M.
(4)
M
\scrZ \ast (M)
is semisimple.
We say that a module M \scrZ \ast -semilocal if M has one of the equal conditions of Lemma 1.1 as
a proper generalization of semilocal modules. In Section 2, we obtain the basic properties of these
modules. We show that the class of \scrZ \ast -semilocal modules is closed under submodules, direct sums
and factor modules. We prove that a ring R is \scrZ \ast -semilocal if and only if every left R-module is
\scrZ \ast -semilocal if and only if every injective left R-module is semilocal. Let \scrR \scrS be the class of all
short exact sequences \BbbE : 0 - \rightarrow M
\psi - \rightarrow N
\phi - \rightarrow K - \rightarrow 0 such that \mathrm{I}\mathrm{m} (\psi ) has a \scrZ \ast -supplement in
N. In Section 3, we show that \scrR \scrS is a proper class over left hereditary rings. We study on some
homological objects of the proper class \scrR \scrS in the same section. In particular, we show that over left
hereditary rings the proper class \scrR \scrS is coinjectively generated by all \mathrm{R}\mathrm{a}\mathrm{d}-small modules.
The following lemma will be frequently used in this paper.
Lemma 1.2 (see [13], Lemma 2.6). The class of \mathrm{R}\mathrm{a}\mathrm{d}-small left R-modules is closed under sub-
modules, direct sums and factor modules.
2. \bfscrZ \ast -semilocal modules and rings. Let M be a module. M is called semilocal if
M
\mathrm{R}\mathrm{a}\mathrm{d}(M)
is semisimple, and a ring R is called semilocal if
R
\mathrm{R}\mathrm{a}\mathrm{d}(R)
is a semisimple ring (see [8]).
It is clear that every semilocal module is \scrZ \ast -semilocal, but the following example shows that the
converse is not true, in general. Firstly, we need the following simple fact.
Lemma 2.1. Every \mathrm{R}\mathrm{a}\mathrm{d}-small module is \scrZ \ast -semilocal.
Proof. Let M be a \mathrm{R}\mathrm{a}\mathrm{d}-small module. Then \scrZ \ast (M) =M. Thus, it is \scrZ \ast -semilocal.
Example 2.1. Let M =\BbbZ \BbbZ . Since M is a small submodule of the injective hull of E(M), it is
\mathrm{R}\mathrm{a}\mathrm{d}-small. So, Z\ast (M) = M. Applying Lemma 2.1, M is Z\ast -semilocal. On the other hand, M is
not semilocal.
Recall from [8] that a module M is weakly supplemented if every submodule U of M has a
weak supplement V in M, that is, M = U + V and U \cap V \ll M. Every supplemented module is
weakly supplemented and weakly supplemented modules are semilocal.
Corollary 2.1. Let M be a module over an arbitrary ring. Suppose that \scrZ \ast (M) is a small
submodule of M. Then the following statements are equivalent:
(1) M is weakly supplemented,
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 3
402 E. TÜRKMEN
(2) M is semilocal,
(3) M is \scrZ \ast -semilocal.
Proof. (1) =\Rightarrow (2) and (2) =\Rightarrow (3) are clear.
(3) =\Rightarrow (1) Let U \subseteq M. By (3), there exists a submodule V of M such that M = U + V and
U \cap V \subseteq \scrZ \ast (M). Since \scrZ \ast (M) is a small submodule of M, it follows from [14] (19.3.(4)) that
U \cap V \ll M. Thus, V is a weak supplement of U in M. Hence, M is weakly supplemented.
Recall that a module M is radical if M = \mathrm{R}\mathrm{a}\mathrm{d}(M), that is, M has no maximal submodules.
Lemma 2.2. Every radical module is \mathrm{R}\mathrm{a}\mathrm{d}-small.
Proof. For a radical module M, let m \in M. Then Rm \ll M. So, Rm is small. Thus,
m \in \scrZ \ast (M).
Let M be a module. By P (M), we denote the sum of all radical submodules of M. P (M) is
the largest radical submodule of M. By using Lemmas 2.1 and 2.2, we obtain the following fact.
Corollary 2.2. P (M) is \scrZ \ast -semilocal for every module M.
It is well known that any submodule of a semilocal module need not be semilocal. For example,
\BbbZ \BbbZ \subseteq \BbbZ \BbbQ . But, we have the following proposition.
Proposition 2.1. Every submodule of a \scrZ \ast -semilocal module is \scrZ \ast -semilocal.
Proof. Let M be a \scrZ \ast -semilocal module and U \subseteq N \subseteq M be submodules. Since M is
\scrZ \ast -semilocal, we can write M = U + V and U \cap V is \mathrm{R}\mathrm{a}\mathrm{d}-small for some submodule V of M.
By using the modular law, N = N \cap M = N \cap (U + V ) = U + (N \cap V ), and U \cap (N \cap V ) =
= (U \cap N) \cap V = U \cap V is \mathrm{R}\mathrm{a}\mathrm{d}-small. Hence, N is \scrZ \ast -semilocal.
Proposition 2.2. Every factor module of a \scrZ \ast -semilocal module is \scrZ \ast -semilocal.
Proof. For a \scrZ \ast -semilocal module M, let N \subseteq U \subseteq M be submodules. Then there exists a
submodule V of M such that M = U+V and U \cap V is \mathrm{R}\mathrm{a}\mathrm{d}-small. Therefore,
M
N
=
U
N
+
V +N
N
.
By using the canonical epimorphism \pi : M - \rightarrow M
N
, we obtain that
\pi (U \cap V ) =
(U \cap V ) +N
N
=
U \cap (V +N)
N
=
U
N
\cap V +N
N
is \mathrm{R}\mathrm{a}\mathrm{d}-small by Lemma 1.2. Hence, the factor module
M
N
is \scrZ \ast -semilocal.
Theorem 2.1. Every direct sum of \scrZ \ast -semilocal modules is \scrZ \ast -semilocal.
Proof. Let \{ Mi\} i\in I be any collection of \scrZ \ast -semilocal modules, where I is any index set. Put
M = \oplus i\in IMi. It follows from [13] (Lemma 2.3) that
M
\scrZ \ast (M)
=
\oplus i\in IMi
\oplus i\in I\scrZ \ast (Mi)
\sim = \oplus i\in I
Mi
\scrZ \ast (Mi)
is semisimple as a direct sum of these semisimple modules
Mi
\scrZ \ast (Mi)
. Therefore, M is \scrZ \ast -semilocal.
Corollary 2.3. Any sum of \scrZ \ast -semilocal submodules of a module M is \scrZ \ast -semilocal.
Proof. Let \{ Ni\} i\in I be the family of \scrZ \ast -semilocal submodules of the module M. Then, we can
write the epimorphism \Psi : \oplus i\in INi - \rightarrow
\sum
i\in I
Ni via \Psi ((ai)i\in I) =
\sum
i\in I0
ai, where I0 is the finite
set of the index set I. By Theorem 2.1, the external direct sum \oplus i\in INi is a \scrZ \ast -semilocal module.
It follows from Proposition 2.2 that the submodule
\sum
i\in I
Ni is \scrZ \ast -semilocal.
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 3
\scrZ \ast -SEMILOCAL MODULES AND THE PROPER CLASS \scrR \scrS 403
Remark 2.1. Let R be a ring with identity. Suppose that RR is a \scrZ \ast -semilocal R-module.
Then, by Lemma 1.1,
R
\scrZ \ast (R)
is a semisimple left R-module. Therefore,
R
\scrZ \ast (R)
is a semisimple
R
\scrZ \ast (R)
-module and so
R
\scrZ \ast (R)
is a semisimple ring. It follows that
R
\scrZ \ast (R)
is a semisimple right
R-module. That is, RR is a \scrZ \ast -semilocal R-module. Similarly, if RR is a \scrZ \ast -semilocal R-module,
it can be shown that RR is a \scrZ \ast -semilocal R-module. By using this fact, we say that R is a
\scrZ \ast -semilocal ring if RR (or RR) is a \scrZ \ast -semilocal R-module.
It is shown in [8] (Theorem 3.5) that a ring R is semilocal if and only if every left R-module is
semilocal. Now, we give an analogue of this fact for \scrZ \ast -semilocal rings.
Lemma 2.3. Let E be an injective module. Then E is \scrZ \ast -semilocal if and only if it is semilocal.
Proof. Let E be a \scrZ \ast -semilocal module and U \subseteq E. Then there exists a submodule V of E such
that E = U + V and U \cap V is \mathrm{R}\mathrm{a}\mathrm{d}-small. Since E is injective, \scrZ \ast (E) = E \cap \mathrm{R}\mathrm{a}\mathrm{d}(E) = \mathrm{R}\mathrm{a}\mathrm{d}(E).
So U \cap V \subseteq \mathrm{R}\mathrm{a}\mathrm{d}(E). Hence, E is semilocal.
Theorem 2.2. The following statements are equivalent for a ring R:
(1) R is \scrZ \ast -semilocal,
(2) every left R-module is \scrZ \ast -semilocal,
(3) every injective left R-module is semilocal.
Proof. (1) =\Rightarrow (2) Let M be any left R-module. Then, for an index set I, there exists an
epimorphism \Psi : R(I) - \rightarrow M. Since R is \scrZ \ast -semilocal, it follows from Theorem 2.1 that the left
free R-module R(I) is \scrZ \ast -semilocal. Therefore, M is \scrZ \ast -semilocal by Proposition 2.2.
(2) =\Rightarrow (3) It is obvious.
(3) =\Rightarrow (2) For any module M, the injective hull E(M) is semilocal. Therefore, E(M) is
\scrZ \ast -semilocal. Applying Proposition 2.1, we deduce that M is \scrZ \ast -semilocal.
(2) =\Rightarrow (1) It follows from (2) that RR is \scrZ \ast -semilocal. Thus, R is a \scrZ \ast -semilocal ring.
In [13], a ring R is called left cosingular if RR is \mathrm{R}\mathrm{a}\mathrm{d}-small. Every commutative domain (which
is not field) is left (right) cosingular. It is proven in [13] (Lemma 2.8) that R is a left cosingular
ring if and only if every injective left R-module is radical. By using this fact, Theorem 2.2 and
Lemma 2.1, we obtain that every left cosingular ring is \scrZ \ast -semilocal. Now, we shall show that a
\scrZ \ast -semilocal ring need not be left cosingular in the following example.
Example 2.2. Let n > 1 be a non-prime positive element of \BbbZ . Then the ring \BbbZ n is
\scrZ \ast -semilocal but not cosingular.
A ring R is called left hereditary if every factor module of an injective left R-module is injective
(see [6]).
Lemma 2.4 (see [7], Theorem 3). Let R be a left hereditary ring and \BbbE : 0 - \rightarrow M
\psi - \rightarrow N
\phi - \rightarrow
\phi - \rightarrow K - \rightarrow 0 be a short exact sequence of left R-modules. Then M and K are small modules if
and only if N is a small module.
We give an analogous characterization of this fact for \mathrm{R}\mathrm{a}\mathrm{d}-small modules.
Lemma 2.5. Let R be a left hereditary ring and \BbbE : 0 - \rightarrow M
f - \rightarrow N
g - \rightarrow K - \rightarrow 0 be a short
exact sequence of left R-modules. Then M and K are \mathrm{R}\mathrm{a}\mathrm{d}-small modules if and only if N is a
\mathrm{R}\mathrm{a}\mathrm{d}-small module.
Proof. (=\Rightarrow ) To simplify the notation, we think of M as a submodule of N. Since M is \mathrm{R}\mathrm{a}\mathrm{d}-
small, we get M \subseteq \mathrm{R}\mathrm{a}\mathrm{d}(E(M)). Therefore, M \subseteq \mathrm{R}\mathrm{a}\mathrm{d}(E(N)). Moreover,
N
M
is \mathrm{R}\mathrm{a}\mathrm{d}-small in
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 3
404 E. TÜRKMEN
E(N)
M
is injective over a left hereditary ring R. Thus, N \subseteq \mathrm{R}\mathrm{a}\mathrm{d}(E(N)). This means that N is
\mathrm{R}\mathrm{a}\mathrm{d}-small.
(\Leftarrow =) It follows from Lemma 1.2.
Lemma 2.6. Let R be a left hereditary ring and M be a left R-module. Suppose that a
submodule N of M is \mathrm{R}\mathrm{a}\mathrm{d}-small. Then \scrZ \ast
\biggl(
M
N
\biggr)
=
\scrZ \ast (M)
N
.
Proof. By the hypothesis, we have N \subseteq \scrZ \ast (M). It follows that
\scrZ \ast (M) +N
N
=
\scrZ \ast (M)
N
\subseteq \scrZ \ast
\biggl(
M
N
\biggr)
.
Let m+N \in \scrZ \ast
\biggl(
M
N
\biggr)
. Then R(m+N) =
Rm+N
N
is a \mathrm{R}\mathrm{a}\mathrm{d}-small module. Now, consider the
following exact sequence:
0 - \rightarrow Rm \cap N i - \rightarrow Rm
\pi - \rightarrow Rm+N
N
- \rightarrow 0,
where i is the canonical injection and \pi is the canonical projection. Applying Lemma 2.5, since R
is left hereditary, Rm is \mathrm{R}\mathrm{a}\mathrm{d}-small and so m \in \scrZ \ast (M). This means that \scrZ \ast
\biggl(
M
N
\biggr)
\subseteq \scrZ \ast (M)
N
.
Hence, \scrZ \ast
\biggl(
M
N
\biggr)
=
\scrZ \ast (M)
N
.
Proposition 2.3. Let R be a left hereditary ring and M be a left R-module. If a submodule N
of M is \mathrm{R}\mathrm{a}\mathrm{d}-small, M is \scrZ \ast -semilocal if and only if
M
N
is \scrZ \ast -semilocal.
Proof. (=\Rightarrow ) By Proposition 2.2.
(\Leftarrow =) Let U \subseteq M be a submodule. By the hypothesis, we can write
M
N
=
U +N
N
+
V
N
and
U +N
N
\cap V
N
is \mathrm{R}\mathrm{a}\mathrm{d}-small for some submodule
V
N
of
M
N
. Then M = U + V. Now,
U +N
N
\cap V
N
=
(U +N) \cap V
N
=
U \cap V +N
N
\subseteq \scrZ \ast
\biggl(
M
N
\biggr)
=
\scrZ \ast (M)
N
according to Lemma 2.6. So, U \cap V \subseteq \scrZ \ast (M). Thus, M is \scrZ \ast -semilocal.
In [14], over an arbitrary ring a module P is said to be a small cover of a module M if there
exists an epimorphism f : P - \rightarrow M with \mathrm{K}\mathrm{e}\mathrm{r} (f) \ll P. A submodule K of M is small in M if
and only if M is a small cover of
M
K
. By using Proposition 2.3, we obtain the following result.
Corollary 2.4. Let R be a left hereditary ring and M be a \scrZ \ast -semilocal R-module. Then every
small cover of M is \scrZ \ast -semilocal.
Proof. Let f : P - \rightarrow M be a small cover. Then \mathrm{K}\mathrm{e}\mathrm{r} (f) is a small submodule of P and
so \mathrm{K}\mathrm{e}\mathrm{r} (f) is \mathrm{R}\mathrm{a}\mathrm{d}-small. Since M is \scrZ \ast -semilocal, we get
P
\mathrm{K}\mathrm{e}\mathrm{r} (f)
is \scrZ \ast -semilocal. Applying
Proposition 2.3, we deduce that P is \scrZ \ast -semilocal.
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 3
\scrZ \ast -SEMILOCAL MODULES AND THE PROPER CLASS \scrR \scrS 405
3. The proper class \scrR \scrS .
Definition 3.1. Let \scrP be a class of short exact sequences of left R-modules and R-module
homomorphisms. If a short exact sequence \BbbE : 0 - \rightarrow M
\psi - \rightarrow N
\phi - \rightarrow K - \rightarrow 0 belongs to \scrP , then \psi
is said to be a \scrP -monomorphism and \phi is said to be an \scrP -epimorphism.
The class \scrP is said to be a proper class (in the sense of Buchsbaum) if it has the following
properties:
(P1) If the short exact sequence \BbbE : 0 - \rightarrow M
\psi - \rightarrow N
\phi - \rightarrow K - \rightarrow 0 is in \scrP , then \scrP contains
every short exact sequence isomorphic to \BbbE .
(P2) \scrP contains all splitting short exact sequences.
(P3) The composite of two \scrP -monomorphisms is a \scrP -monomorphism if this composite is de-
fined.
(P\prime
3) The composite of two \scrP -epimorphisms is a \scrP -epimorphism if this composite is defined.
(P4) If \psi 1, \psi 2 are monomorphisms and \psi 2\psi 1 is a \scrP -monomorphism, then \psi 1 is a \scrP -mono-
morphism.
(P\prime
4) If \phi 1, \phi 2 are epimorphisms and \phi 2\phi 1 is an \scrP -epimorphism, then \phi 2 is an \scrP -epimorphism.
Example 3.1. We list some examples of proper classes:
(1) The smallest proper class \scrS plit of all splitting short exact sequences of left R-modules.
(2) The largest proper class \scrA bs of all short exact sequences of left R-modules.
(3) The proper class \scrS upp of all short exact sequences 0 - \rightarrow M
\psi - \rightarrow N
\phi - \rightarrow K - \rightarrow 0 such that
\mathrm{I}\mathrm{m}(\psi ) is a supplement of some submodule of N (see [5]).
(4) The proper class \scrC o-\scrN eat of all short exact sequences 0 - \rightarrow M
\psi - \rightarrow N
\phi - \rightarrow K - \rightarrow 0 such
that \mathrm{I}\mathrm{m}(\psi ) is a \mathrm{R}\mathrm{a}\mathrm{d}-supplement of some submodule of N (see [10]).
(5) Over left hereditary rings the proper class \scrS \scrS of all short exact sequences 0 - \rightarrow M
\psi - \rightarrow
\psi - \rightarrow N
\phi - \rightarrow K - \rightarrow 0 such that \mathrm{I}\mathrm{m} (\psi ) has a small supplement in N, that is, N = \mathrm{I}\mathrm{m}(\psi ) + V and
\mathrm{I}\mathrm{m}(\psi ) \cap V is a small module (see [1]).
Now, we have the following implications on the the above classes of left R-modules:
\scrS plit \subseteq \scrS upp \subseteq \scrC o-\scrN eat \subseteq \scrA bs and \scrS plit \subseteq \scrS upp \subseteq \scrS \scrS \subseteq \scrA bs.
Let \scrR \scrS be the class of all short exact sequences \BbbE : 0 - \rightarrow M
\psi - \rightarrow N
\phi - \rightarrow K - \rightarrow 0 such that
\mathrm{I}\mathrm{m}(\psi ) has a \scrZ \ast -supplement in N, that is, \mathrm{I}\mathrm{m}(\psi ) + V = N and \mathrm{I}\mathrm{m}(\psi ) \cap V is \mathrm{R}\mathrm{a}\mathrm{d}-small for some
submodule V of N. It is obvious that \scrC o-\scrN eat \subseteq \scrR \scrS and \scrS \scrS \subseteq \scrR \scrS . The following example
shows that \scrR \scrS contains properly the class \scrS \scrS and the class \scrC o-Neat.
Example 3.2. (1) Let R be a local Dedekind domain (i.e., DVR) with quotient K \not = R
\Bigl(
e.g.,
the ring \BbbZ (p) containing all rational numbers of the form
a
b
with p \nmid b for any prime p in \BbbZ
\Bigr)
.
Put N = R(\BbbN ) and M = \mathrm{R}\mathrm{a}\mathrm{d}(N). Consider the extension \BbbE : 0 - \rightarrow M
\iota - \rightarrow N
\pi - \rightarrow K - \rightarrow 0,
where K =
N
M
. Then \BbbE is an element of \scrR \scrS . Howewer, it is not in \scrS \scrS because M has no (weak)
supplements in the projective module N.
(2) Let N =\BbbZ \BbbZ and M =\BbbZ 2\BbbZ . Put K =\BbbZ
\biggl(
\BbbZ
2\BbbZ
\biggr)
. Then the extension \BbbE : 0 - \rightarrow M
\iota - \rightarrow
\iota - \rightarrow N
\pi - \rightarrow K - \rightarrow 0 is in the class \scrR \scrS by Theorem 2.2. On the other hand, \BbbE is not an element of
\scrC o-Neat since M is not \mathrm{R}\mathrm{a}\mathrm{d}-supplement in N.
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406 E. TÜRKMEN
Proposition 3.1. Let R be an arbitrary ring. If every injective left R-module has a small
radical, then \scrR \scrS = \scrS \scrS .
Proof. Let \BbbE : 0 - \rightarrow M
\psi - \rightarrow N
\phi - \rightarrow K - \rightarrow 0 be any element of the class \scrR \scrS . Then there
exists a submodule V of N such that N =M + V and M \cap V is \mathrm{R}\mathrm{a}\mathrm{d}-small. Therefore, M \cap V \subseteq
\subseteq \mathrm{R}\mathrm{a}\mathrm{d}(E(M \cap V )). By the assumption and [6] (11.5.5, \S 11.6.3), we obtain that M \cap V is a small
submodule of the injective hull E(M \cap V ). It means that M has a small supplement in N. Hence,
\scrR \scrS = \scrS \scrS .
A ring R is said to be a left max ring if every non-zero left R-module has a maximal submodule.
Now we have the following:
Corollary 3.1. Let R be a left max ring. Then \scrR \scrS = \scrS \scrS .
Proof. Since R is left max, every left R-module has a small radical. Hence, the proof follows
from Proposition 3.1.
Example 3.3. Consider the non-Noetherian commutative ring which is the direct product\prod \infty
i\geq 1
Fi, where Fi = F is any field. Suppose that R is the subring of the ring consisting of
all sequences (rn)n\in \BbbN such that there exist r \in F, m \in \BbbN with rn = r for all n \geq m. Let N =R R.
Then N is a regular module which is not semisimple. Put M = \mathrm{S}\mathrm{o}\mathrm{c} (N) and K =
N
M
. Then the
extension \BbbE : 0 - \rightarrow M
\iota - \rightarrow N
\pi - \rightarrow K - \rightarrow 0 is not in \scrR \scrS .
Theorem 3.1. A ring R is a Z\ast -semilocal ring if and only if \scrR \scrS = \scrA bs.
Proof. By Theorem 2.2.
Observe from Theorem 3.1 that over \scrZ \ast -semilocal rings (in particular, semilocal rings or com-
mutative domains), \scrR \scrS is a proper class.
The following the structure of the Abelian group \mathrm{E}\mathrm{x}\mathrm{t}R(K,M) is given in the book [9, p. 63 – 71],
and we recall them for the convenience of the reader:
Let R be an arbitrary ring with identity and \BbbE : 0 - \rightarrow M
\psi - \rightarrow N
\phi - \rightarrow K - \rightarrow 0 be a short exact
sequence of left R-modules and module homomorphisms. Then \BbbE is called an extension of M by K.
By \mathrm{E}\mathrm{x}\mathrm{t}R(K,M) we will denote the set of all equivalence classes of extensions of M by K. Let \BbbE 1 :
0 - \rightarrow M
\psi 1 - \rightarrow N1
\phi 1 - \rightarrow K - \rightarrow 0 and \BbbE 2 : 0 - \rightarrow M
\psi 2 - \rightarrow N2
\phi 2 - \rightarrow K - \rightarrow 0 be any elements of
\mathrm{E}\mathrm{x}\mathrm{t}R(K,M). We define the direct sum of \BbbE 1 and \BbbE 2 as follows:
\BbbE 1 \oplus \BbbE 2 : 0 - \rightarrow M \oplus M
\psi - \rightarrow N1 \oplus N2
\phi - \rightarrow K \oplus K - \rightarrow 0,
where \psi (m1,m2) = (\psi 1 \oplus \psi 2)(m1,m2) = (\psi 1(m1), \psi 2(m2)) for all (m1,m2) \in M \oplus M and
\phi (n1, n2) = (\phi 1 \oplus \phi 2)(n1, n2) = (\phi 1(n1), \phi 2(n2)) for all (n1, n2) \in N1 \oplus N2. Then \BbbE 1 \oplus \BbbE 2 is
a short exact sequence. The Baer sum of \BbbE 1 and \BbbE 2, \BbbE 1 + \BbbE 2 = \bigtriangledown M (\BbbE 1 \oplus \BbbE 2)\bigtriangleup K , where the
diagonal map \bigtriangleup K(k) = (k, k) for all k \in K and the codiagonal map \bigtriangledown M (m1,m2) = m1+m2 for
all (m1,m2) \in M \oplus M. Therefore, \mathrm{E}\mathrm{x}\mathrm{t}R(K,M) is an Abelian group under Baer sum of extensions.
Note that the split extension 0 - \rightarrow M - \rightarrow M \oplus K - \rightarrow K - \rightarrow 0 is the zero element of this group
and the inverse of an extension \BbbE : 0 - \rightarrow M
\psi - \rightarrow N
\phi - \rightarrow K - \rightarrow 0 is the extension ( - IM )\BbbE .
The set \mathrm{E}\mathrm{x}\mathrm{t}\scrP (K,M) of all short exact sequences of \mathrm{E}\mathrm{x}\mathrm{t}R(K,M) that belongs to a proper class
\scrP is a subgroup of the group of \mathrm{E}\mathrm{x}\mathrm{t}R(K,M).
Theorem 3.2 (see [12], Theorem 1.1). Let \scrP be a class of short exact sequences for left
R-modules. If \mathrm{E}\mathrm{x}\mathrm{t}\scrP (K,M) is a subfunctor of \mathrm{E}\mathrm{x}\mathrm{t}R(K,M), \mathrm{E}\mathrm{x}\mathrm{t}\scrP (K,M) is a subgroup of
\mathrm{E}\mathrm{x}\mathrm{t}R(K,M) for every R-modules M, K and the composition of two \scrP -monomorphism
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\scrZ \ast -SEMILOCAL MODULES AND THE PROPER CLASS \scrR \scrS 407
(or \scrP -epimorphisms) is a \scrP -monomorphism (an \scrP -epimorphism, respectively), then \scrP is a proper
class.
Using Theorem 3.2, we shall prove that \scrR \scrS is a proper class over left hereditary rings.
Lemma 3.1. Let f : M - \rightarrow M \prime be any homomorphism of left R-modules. Then
f\ast : \mathrm{E}\mathrm{x}\mathrm{t}R(K,M) - \rightarrow \mathrm{E}\mathrm{x}\mathrm{t}R
\bigl(
K,M \prime \bigr)
preserves the elements of the class \scrR \scrS .
Proof. Let \BbbE : 0 - \rightarrow M
\psi - \rightarrow N
\phi - \rightarrow K - \rightarrow 0 be any element of \scrR \scrS . Take the left R-module
N \prime =
M \prime \oplus N
H
, where H =
\bigl\{
( - f(m), \psi (m)) \in M \prime \oplus N | m \in M
\bigr\}
is a submodule of M \prime \oplus N.
Define these homomorphisms of left R-modules \psi \prime : M \prime - \rightarrow N \prime via \psi \prime (m\prime ) = (m\prime , 0) + H, \phi \prime :
N \prime - \rightarrow K via \phi \prime ((m\prime , n)) = \phi (n) and h : N - \rightarrow N \prime via h(n) = (0, n) +H. Then f\ast (\BbbE ) = f\BbbE :
0 - \rightarrow M \prime \psi \prime
- \rightarrow N \prime \phi \prime - \rightarrow K - \rightarrow 0 \in \mathrm{E}\mathrm{x}\mathrm{t}R (K,M \prime ) and we obtain the following commutative diagram
with exact rows:
\BbbE : 0 //M
\psi
//
f
��
N
\phi
//
h
��
K // 0
f\BbbE : 0 //M \prime \psi \prime
// N \prime \phi \prime
// K // 0
that is, \psi \prime f = h\psi and \phi \prime h = \phi . Since the extension \BbbE : 0 - \rightarrow M
\psi - \rightarrow N
\phi - \rightarrow K - \rightarrow 0 is in the class
\scrR \scrS of left R-modules, there exists a submodule V of N such that N = \mathrm{I}\mathrm{m} (\psi )+V and \mathrm{I}\mathrm{m} (\psi )\cap V
is Rad-small. By using the above commutative diagram, we obtain that N \prime = \mathrm{I}\mathrm{m} (\psi \prime )+ \mathrm{I}\mathrm{m} (h) and
\mathrm{I}\mathrm{m} (h)\cap \mathrm{I}\mathrm{m} (\psi \prime ) = h(\mathrm{I}\mathrm{m} (\psi )\cap V ). It follows from Lemma 1.2 that \mathrm{I}\mathrm{m} (h)\cap \mathrm{I}\mathrm{m} (\psi \prime ) is \mathrm{R}\mathrm{a}\mathrm{d}-small
as a homomorphic image of the \mathrm{R}\mathrm{a}\mathrm{d}-small module \mathrm{I}\mathrm{m} (\psi ) \cap V. So \mathrm{I}\mathrm{m} (h) is a \scrZ \ast -supplement of
\mathrm{I}\mathrm{m} (\psi \prime ) in N \prime . Thus, f\BbbE = f\ast (\BbbE ) \in \scrR \scrS .
Observe from Lemma 3.1 that if, for all modules M and K, \BbbE \in \mathrm{E}\mathrm{x}\mathrm{t}\scrR \scrS (K,M), then the inverse
extension ( - IM )\BbbE \in \mathrm{E}\mathrm{x}\mathrm{t}\scrR \scrS (K,M).
Lemma 3.2. Let g : K \prime - \rightarrow K be any homomorphism of left R-modules. Then
g\ast : \mathrm{E}\mathrm{x}\mathrm{t}R(K,M) - \rightarrow \mathrm{E}\mathrm{x}\mathrm{t}R
\bigl(
K \prime ,M
\bigr)
preserves the elements of the class \scrR \scrS .
Proof. Let \BbbE : 0 - \rightarrow M
\psi - \rightarrow N
\phi - \rightarrow K - \rightarrow 0 be a short exact sequence in \scrR \scrS . Consider the
left R-submodule N \prime = \{ (n, k\prime ) \in N \oplus K \prime | \phi (n) = g (k\prime )\} of the left R-module N \oplus K \prime . Define
these homomorphisms \phi \prime : N \prime - \rightarrow K \prime via \phi \prime (n, k\prime ) = k\prime , h : N \prime - \rightarrow N via h (n, k\prime ) = n and \psi \prime :
M - \rightarrow N \prime via \psi \prime (m) = (\psi (m), 0). Then we can write the following commutative diagram with
rows:
\BbbE g : 0 //M
\psi \prime
// N \prime \phi \prime
//
h
��
K \prime //
g
��
0
\BbbE : 0 //M
\psi
// N
\phi
// K // 0
where g\ast (\BbbE ) = \BbbE g. Since \BbbE is an element of \scrR \scrS , there exists a submodule V of N such that
N = \mathrm{I}\mathrm{m} (\psi ) + V = and \mathrm{I}\mathrm{m} (\psi ) \cap V is \mathrm{R}\mathrm{a}\mathrm{d}-small. To show N \prime = \mathrm{I}\mathrm{m} (\psi \prime ) + h - 1(V ), let a
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408 E. TÜRKMEN
be any element of N \prime . Then we can write h(a) = \psi (m) + v where m \in M and v \in V. Since
\psi (m) = (h\psi \prime ) (m), we have a - \psi \prime (m) \in h - 1(V ) and this implies that N \prime = \mathrm{I}\mathrm{m} (\psi \prime ) + h - 1(V ).
Let (n, k\prime ) be any element of \mathrm{I}\mathrm{m} (\psi \prime ) \cap h - 1(V ). Since \mathrm{I}\mathrm{m} (\psi \prime ) = \mathrm{K}\mathrm{e}\mathrm{r} (\phi \prime ) , we obtain that
\phi \prime (n, k\prime ) = k\prime = 0. Then g (k\prime ) = \phi (n) = 0, that is, n \in \mathrm{K}\mathrm{e}\mathrm{r} (\phi ). Therefore, n \in \mathrm{I}\mathrm{m} (\psi ) because
\mathrm{I}\mathrm{m} (\psi ) = \mathrm{K}\mathrm{e}\mathrm{r} (\phi ). It follows that n \in \mathrm{I}\mathrm{m} (\psi ) \cap V. Since \mathrm{I}\mathrm{m} (\psi ) \cap V is \mathrm{R}\mathrm{a}\mathrm{d}-small, the module
Rn is small, and so R (n, k\prime ) is small. Thus, \mathrm{I}\mathrm{m} (\psi \prime ) \cap h - 1(V ) is \mathrm{R}\mathrm{a}\mathrm{d}-small. Hence, h - 1(V ) is a
\scrZ \ast -supplement of \mathrm{I}\mathrm{m} (\psi \prime ) in N \prime .
Lemma 3.3. If \BbbE 1,\BbbE 2 \in \mathrm{E}\mathrm{x}\mathrm{t}RS(K,M), then \BbbE 1 \oplus \BbbE 2 \in \mathrm{E}\mathrm{x}\mathrm{t}RS(K \oplus K,M \oplus M).
Proof. Let \BbbE 1 : 0 - \rightarrow M
\psi 1 - \rightarrow N1
\phi 1 - \rightarrow K - \rightarrow 0 and \BbbE 2 : 0 - \rightarrow M
\psi 2 - \rightarrow N2
\phi 2 - \rightarrow K - \rightarrow 0
be two elements of \mathrm{E}\mathrm{x}\mathrm{t}\scrR \scrS (K,M). Then, for i = 1, 2, Ni = M + Vi and M \cap Vi is \mathrm{R}\mathrm{a}\mathrm{d}-small
for some submodules Vi of Ni. Since (M \oplus M) + (V1 \oplus V2) = N1 \oplus N2 and (M \oplus M) \cap (V1 \oplus
\oplus V2) = (M \cap V1)\oplus (M \cap V2), it follows from Lemma 1.2 that the short exact sequence \BbbE 1 \oplus \BbbE 2 :
0 - \rightarrow M \oplus M
\psi - \rightarrow N1 \oplus N2
\phi - \rightarrow K \oplus K - \rightarrow 0 is in \mathrm{E}\mathrm{x}\mathrm{t}RS(K \oplus K,M \oplus M), where \psi = \psi 1 \oplus \psi 2
and \phi = \phi 1 \oplus \phi 2.
Corollary 3.2. \mathrm{E}\mathrm{x}\mathrm{t}\scrR \scrS (K,M) is a subgroup of the extension \mathrm{E}\mathrm{x}\mathrm{t}R(K,M) for every module K
and M. Moreover, \mathrm{E}\mathrm{x}\mathrm{t}\scrR \scrS (K,M) is a subfunctor of the functor \mathrm{E}\mathrm{x}\mathrm{t}R(K,M).
Proof. Let \BbbE 1 : 0 - \rightarrow M
\psi 1 - \rightarrow N1
\phi 1 - \rightarrow K - \rightarrow 0 and \BbbE 2 : 0 - \rightarrow M
\psi 2 - \rightarrow N2
\phi 2 - \rightarrow K - \rightarrow 0
be any elements of \mathrm{E}\mathrm{x}\mathrm{t}\scrR \scrS (K,M). It follows from Lemmas 3.1, 3.2 and 3.3 that the Baer sum
\BbbE 1 + \BbbE 2 of these extensions \BbbE 1 and \BbbE 2 is in \mathrm{E}\mathrm{x}\mathrm{t}\scrR \scrS (K,M). Hence, \mathrm{E}\mathrm{x}\mathrm{t}\scrR \scrS (K,M) is a subgroup
of \mathrm{E}\mathrm{x}\mathrm{t}R(K,M).
Theorem 3.3. Let R be a left hereditary ring. Then \scrR \scrS is a proper class.
Proof. By Theorem 3.2 and Corollary 3.2, it suffices to show that the composition of two
\scrR \scrS -epimorphisms is an \scrR \scrS -epimorphism. Let f : N - \rightarrow N \prime and g : N \prime - \rightarrow K be \scrR \scrS -epimor-
phisms. Now we have the following commutative diagram with exact rows and columns:
0
��
0
��
0 // \mathrm{K}\mathrm{e}\mathrm{r}(f)
i\mathrm{K}\mathrm{e}\mathrm{r}(f)
//M //
��
\mathrm{K}\mathrm{e}\mathrm{r}(g) //
i\mathrm{K}\mathrm{e}\mathrm{r}(g)
��
0
0 // \mathrm{K}\mathrm{e}\mathrm{r}(f) // N
f
//
��
N \prime //
g
��
0
K
��
K
��
0 0
where i\mathrm{K}\mathrm{e}\mathrm{r} (f) and i\mathrm{K}\mathrm{e}\mathrm{r} (g) are the canonical inclusions. By the hypothesis, we can write N =
= \mathrm{K}\mathrm{e}\mathrm{r} (f)+V and \mathrm{K}\mathrm{e}\mathrm{r} (f)\cap V is \mathrm{R}\mathrm{a}\mathrm{d}-small for some submodule V of N, and
N
\mathrm{K}\mathrm{e}\mathrm{r} (f)
=
M
\mathrm{K}\mathrm{e}\mathrm{r} (f)
+
+
L
\mathrm{K}\mathrm{e}\mathrm{r} (f)
and
M \cap L
\mathrm{K}\mathrm{e}\mathrm{r} (f)
is \mathrm{R}\mathrm{a}\mathrm{d}-small for some submodule
L
\mathrm{K}\mathrm{e}\mathrm{r} (f)
of
N
\mathrm{K}\mathrm{e}\mathrm{r} (f)
. Therefore, M =
= M \cap N = M \cap (\mathrm{K}\mathrm{e}\mathrm{r} (f) + V ) = \mathrm{K}\mathrm{e}\mathrm{r} (f) + M \cap V, M \cap L = \mathrm{K}\mathrm{e}\mathrm{r} (f) + M \cap V \cap L and
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\scrZ \ast -SEMILOCAL MODULES AND THE PROPER CLASS \scrR \scrS 409
L = \mathrm{K}\mathrm{e}\mathrm{r} (f) + L \cap V. It follows that N = M + (V \cap L). Applying Lemma 2.5, we deduce that
M \cap V \cap L is \mathrm{R}\mathrm{a}\mathrm{d}-small. This means that the composition gf is an \scrR \scrS -epimorphism.
Let \scrM be a class of short exact sequences. The smallest proper class containing \scrM is said to
be generated by \scrM and denoted by \langle \scrM \rangle . Since the intersection of any proper classes is proper, we
have \langle \scrM \rangle = \cap \{ \scrP | \scrM \subseteq \scrP and \scrP is proper class\} .
By \scrR \scrS mall, we will denote the class of all short exact sequences \BbbE : 0 - \rightarrow M
\psi - \rightarrow N
\phi - \rightarrow
\phi - \rightarrow K - \rightarrow 0 such that \mathrm{I}\mathrm{m}(\psi ) \subseteq \mathrm{R}\mathrm{a}\mathrm{d}(N), and by \scrW \scrR \scrS we will denote the class of all short exact
sequences \BbbE : 0 - \rightarrow M
\psi - \rightarrow N
\phi - \rightarrow K - \rightarrow 0 such that \mathrm{I}\mathrm{m} (\psi ) has a weak \mathrm{R}\mathrm{a}\mathrm{d}-supplement in N,
that is, \mathrm{I}\mathrm{m} (\psi ) + V = N and \mathrm{I}\mathrm{m} (\psi ) \cap V \subseteq \mathrm{R}\mathrm{a}\mathrm{d}(N) for some submodule V of N.
Clearly, \scrR \scrS mall \subseteq \scrW \scrR \scrS \subseteq \scrR \scrS , and so \langle \scrR \scrS mall\rangle \subseteq \langle \scrW \scrR \scrS \rangle \subseteq \scrR \scrS whenever \scrR \scrS is a
proper class. Motivated by [1] (Corollary 3.13), we shall prove that \langle \scrR \scrS mall\rangle = \langle \scrW \scrR \scrS \rangle = \scrR \scrS in
the following theorem.
Theorem 3.4. For the proper class \scrR \scrS , \langle \scrR \scrS mall\rangle = \langle \scrW \scrR \scrS \rangle = \scrR \scrS .
Proof. Let \BbbE : 0 - \rightarrow M
\psi - \rightarrow N
\phi - \rightarrow K - \rightarrow 0 be any element of the proper class \scrR \scrS .
Then, for some submodule V of N, we can write N = M + V and M \cap V is \mathrm{R}\mathrm{a}\mathrm{d}-small. Put
L =M \cap V. Therefore, the extension \BbbE : 0 - \rightarrow M
L
\iota - \rightarrow N
L
\Phi - \rightarrow K - \rightarrow 0 is in the class \langle \scrR \scrS mall\rangle ,
where \iota is the canonical injection, \pi : N - \rightarrow N
L
is the canonical projection and \phi = \Phi \pi . Since
\pi and \Phi are \langle \scrR \scrS mall\rangle -epimorphisms, we get that \phi is \langle \scrR \scrS mall\rangle -epimorphism. It means that
\scrR \scrS \subseteq \langle \scrR \scrS mall\rangle .
Let \scrP be a proper class. A module M is said to be \scrP -injective (respectively, \scrP -coinjective) if the
subgroup \mathrm{E}\mathrm{x}\mathrm{t}\scrP (K,M) = 0 (respectively, \mathrm{E}\mathrm{x}\mathrm{t}\scrP (K,M) = \mathrm{E}\mathrm{x}\mathrm{t}R(K,M)) for all left R-modules K.
Now we prove that weak \mathrm{R}\mathrm{a}\mathrm{d}-supplement submodules of \scrR \scrS -coinjective modules are \scrR \scrS -
coinjective.
Proposition 3.2. Let R be a left hereditary ring and M be a \scrR \scrS -coinjective R-module. Then
every weak \mathrm{R}\mathrm{a}\mathrm{d}-supplement submodule of M is \scrR \scrS -coinjective.
Proof. Let A be a weak \mathrm{R}\mathrm{a}\mathrm{d}-supplement submodule of M. Then the extension \BbbE : 0 - \rightarrow A
\iota - \rightarrow
\iota - \rightarrow M
\pi - \rightarrow M
A
- \rightarrow 0 is an element of the class \scrW \scrR \scrS and so it is in \scrR \scrS . Hence, by [11]
(Proposition 1.8), A is \scrR \scrS -coinjective.
Now we characterize \scrR \scrS -coinjective modules via weak \mathrm{R}\mathrm{a}\mathrm{d}-supplements in the following the-
orem which is adapted of [3] (Theorem 4.1).
Theorem 3.5. For a module M over a left hereditary ring R, the following statements are
equivalent:
(1) M is \scrR \scrS -coinjective,
(2) M has a weak \mathrm{R}\mathrm{a}\mathrm{d}-supplement in E(M).
Proof. (1) =\Rightarrow (2) Let \delta : M - \rightarrow E(M) be the essential monomorphism. Without loss of
generality, we take M \subseteq E(M). By (1), there exists a submodule V of E(M) such that M + V =
= E(M) and M \cap V is \mathrm{R}\mathrm{a}\mathrm{d}-small. Since E(M) is injective, \scrZ \ast (E(M)) = \mathrm{R}\mathrm{a}\mathrm{d}(E(M)), and so
M \cap V \subseteq \mathrm{R}\mathrm{a}\mathrm{d}(E(M)). Thus, V is a weak Rad-supplement of M in E(M).
(2) =\Rightarrow (1) is clear by [11] (Proposition 1.7).
Corollary 3.3. Let R be a left hereditary ring. Then RR is \scrR \scrS -coinjective if and only if there
exists a submodule S of E(RR) such that E(RR) = R+ S and R \cap S \subseteq \mathrm{R}\mathrm{a}\mathrm{d}(E(RR)).
The following fact is a direct consequence of Theorem 3.5.
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410 E. TÜRKMEN
Corollary 3.4. Every \mathrm{R}\mathrm{a}\mathrm{d}-small module over a left hereditary ring is \scrR \scrS -coinjective.
Proof. Let M be a \mathrm{R}\mathrm{a}\mathrm{d}-small module. Then, M \subseteq \mathrm{R}\mathrm{a}\mathrm{d}(E(M)). Therefore, E(M) = E(M)+
+M and M \cap E(M) \subseteq \mathrm{R}\mathrm{a}\mathrm{d}(E(M)). So E(M) is a weak \mathrm{R}\mathrm{a}\mathrm{d}-supplement of M in the injective
hull E(M). Hence, M is \scrR \scrS -coinjective by Theorem 3.5.
The smallest proper class for which every module from the class of modules \scrM is coinjective is
denoted by k(\scrM ). Such classes are said to be coinjectively generated by \scrM .
Proposition 3.3. Let R be a left hereditary ring. The proper class \scrR \scrS is coinjectively generated
by all \mathrm{R}\mathrm{a}\mathrm{d}-small left R-modules.
Proof. We shall show that \scrR \scrS = k(\scrR \scrS mall). It follows from Corollary 3.4 that every \mathrm{R}\mathrm{a}\mathrm{d}-
small R-module is \scrR \scrS -coinjective, and so k(\scrR \scrS mall) \subseteq \scrR \scrS . By Proposition 3.2, we get \scrR \scrS =
= \langle \scrR \scrS mall\rangle \subseteq k(\scrR \scrS mall). Hence, \scrR \scrS = k(\scrR \scrS mall).
Let \scrP be a proper class. The global dimension of \scrP is defined as
gl.dim \scrP =
\bigl\{
n | \mathrm{E}\mathrm{x}\mathrm{t}n+1
\scrP (K,M) = 0 for all M and K left R-modules
\bigr\}
.
If there no such n, then gl.dim \scrP = \infty .
Theorem 3.6. \mathrm{g}\mathrm{l}.\mathrm{d}\mathrm{i}\mathrm{m} \scrR \scrS \leq 1.
Proof. It follows from Theorem 3.3 and [2].
Recall that a ring R is said to be a left V-ring if every simple left R-module is injective. The
following next theorem characterizes the left hereditary rings in which \scrR \scrS -coinjective modules are
injective.
Theorem 3.7. The following statements are equivalent for a left hereditary ring R:
(1) every \scrR \scrS -coinjective module is injective,
(2) every \mathrm{R}\mathrm{a}\mathrm{d}-small module is injective,
(3) every small module is injective,
(4) R is a left V-ring.
Proof. (1) =\Rightarrow (2) If follows from Corollary 3.4.
(2) =\Rightarrow (3) Since small modules are \mathrm{R}\mathrm{a}\mathrm{d}-small.
(3) =\Rightarrow (4) By [14] (23.1), it suffices to prove that, for any left R-module M, \mathrm{R}\mathrm{a}\mathrm{d}(M) = 0.
Let m \in \mathrm{R}\mathrm{a}\mathrm{d}(M). Then Rm is a small submodule of M. By (3), we can write the decomposition
M = Rm \oplus K for some submodule K of M. It follows that m = 0. Hence, we obtain that
\mathrm{R}\mathrm{a}\mathrm{d}(M) = 0.
(4) =\Rightarrow (1) Let M be a \scrR \scrS -coinjective module and N be any extension of M. Then N =
= M + V and M \cap V \subseteq \mathrm{R}\mathrm{a}\mathrm{d}(E(M \cap V )) for some submodule V of N. Since R is a left V -ring,
by [14] (23.1), we get M \cap V \subseteq \mathrm{R}\mathrm{a}\mathrm{d}(E(M \cap V )) = 0. Thus, M is a direct summand of N.
It means that M is injective.
Let \scrP be a proper class. A module M is said to be \scrP -projective (respectively, \scrP -coprojective)
if the subgroup \mathrm{E}\mathrm{x}\mathrm{t}\scrP (M,K) = 0 (respectively, \mathrm{E}\mathrm{x}\mathrm{t}\scrP (M,K) = \mathrm{E}\mathrm{x}\mathrm{t}R(M,K)) for all left R-modu-
le K.
Theorem 3.8. Let M be a module over a left hereditary ring. Then, the following statements
are equivalent:
(1) M is \scrR \scrS -projective.
(2) \mathrm{E}\mathrm{x}\mathrm{t}R(M,K) = 0 for every \mathrm{R}\mathrm{a}\mathrm{d}-small module K.
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 3
\scrZ \ast -SEMILOCAL MODULES AND THE PROPER CLASS \scrR \scrS 411
Proof. (1) =\Rightarrow (2) is clear.
(2) =\Rightarrow (1) Let 0 - \rightarrow A
\psi - \rightarrow B
\phi - \rightarrow C - \rightarrow 0 be any element of \scrR \scrS . So, B = A + D and
A \cap D is \mathrm{R}\mathrm{a}\mathrm{d}-small for some submodule D of B. Put F = A \cap D. Then the short exact sequence
0 - \rightarrow A
F
i1 - \rightarrow B
F
\pi 1 - \rightarrow D
F
- \rightarrow 0 splits, where i1 is the canonical injection and \pi 1 is the canonical
projection. Now we can write the following commutative diagram:
0 // A
\psi
//
\pi F
��
B
\phi
//
\pi
��
C //
IC
��
0
0 //
A
F
i1 //
B
F
f\pi 1
// C // 0
where \pi F and \pi are canonical projections. Applying the functor \mathrm{H}\mathrm{o}\mathrm{m}(M, .), we get
\mathrm{H}\mathrm{o}\mathrm{m}(M,B)
\phi \ast
//
\pi \ast
��
\mathrm{H}\mathrm{o}\mathrm{m}(M,C) // 0
\mathrm{H}\mathrm{o}\mathrm{m}
\biggl(
M,
B
F
\biggr)
(f\pi 1)\ast
//
��
\mathrm{H}\mathrm{o}\mathrm{m}(M,C) // 0
\mathrm{E}\mathrm{x}\mathrm{t}R(M,F )
Then (f\pi 1)\ast is an epimorphism. It follows from (2) that \mathrm{E}\mathrm{x}\mathrm{t}R(M,F ) = 0. So \pi \ast is an epimor-
phism. This means that \phi \ast is an epimorphism. Consequently, M is \scrR \scrS -projective.
References
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Received 01.06.16,
after revision — 09.02.17
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 3
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| id | umjimathkievua-article-1447 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:05:33Z |
| publishDate | 2019 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/4b/6789db85b25d61b0fad41f8f6a18294b.pdf |
| spelling | umjimathkievua-article-14472019-12-05T08:55:13Z $\scr{Z^{ \ast}}$ - semilocal modules and the proper class $\scr{RS}$ $\scr{Z^{ \ast}}$ - напiвлокальнi модулi та власний клас $\scr{RS}$ Türkmen, E. Тюркмен, Є. Over an arbitrary ring, a module $M$ is said to be $\scr{Z^{ \ast}}$ -semilocal if every submodule $U$ of $M$ has a $\scr{Z^{ \ast}}$ -supplement $V$ in $M$, i.e., $M = U + V$ and $U \cap V \subseteq \scr{Z^{ \ast}} (V )$, where $\scr{Z^{ \ast}}(V ) = \{m \in V | Rm$ is a small module $\}$ is the $\mathrm{R}\mathrm{a}\mathrm{d}$-small submodule. In this paper, we study basic properties of these modules as a proper generalization of semilocal modules. In particular, we show that the class of $\scr{Z^{ \ast}}$ -semilocal modules is closed under submodules, direct sums, and factor modules. Moreover, we prove that a ring $R$ is $\scr{Z^{ \ast}}$ -semilocal if and only if every injective left R-module is semilocal. In addition, we show that the class $\scr{RS}$ of all short exact sequences $E :0 \xrightarrow{\psi} M \xrightarrow{\phi} K \rightarrow 0$ such that $\mathrm{I}\mathrm{m}(\psi )$ has a $\scr{Z^{ \ast}}$ -supplement in $N$ is a proper class over left hereditary rings. We also study some homological objects of the proper class $\scr{RS}$ . Над довiльним кiльцем модуль $M$ називається $\scr{Z^{ \ast}}$ -напiвлокальним, якщо кожний пiдмодуль $U$ модуля $M$ має $\scr{Z^{\ast}}$ -доповнення $V$ в $M$, тобто $M = U + V$ i $U \cap V \subseteq \scr{Z^{\ast}} (V)$, де $\scr{Z^{\ast}}(V) = \{ m \in V | Rm$ — малий модуль$\}$ — $\mathrm{R}\mathrm{a}\mathrm{d}$-малий пiдмодуль. У цiй роботi вивчаються базовi властивостi таких модулiв, як вiдповiдного узагальнення напiвлокальних модулiв. Зокрема, показано, що клас $\scr{Z^{ \ast}}$ -напiвлокальних модулiв є замкненим вiдносно пiдмодулiв, прямих сум i фактор-модулiв. Крiм того, доведено, що кiльце $R \in \scr{Z^{ \ast}}$ -напiвлокальним тодi i тiльки тодi, коли кожен iн’єктивний лiвий $R$-модуль є напiвлокальним. Також встановлено, що клас $\scr{RS}$ усiх коротких послiдовностей $E :0 \xrightarrow{\psi} M \xrightarrow{\phi} K \rightarrow 0$ таких, що $\mathrm{Im}(\psi)$ має $\scr{Z^{ \ast}}$-доповнення в $N$, є власним класом над лiвими спадковими кiльцями. Вивчено також деякi гомологiчнi об’єкти власного класу $\scr{RS}$. Institute of Mathematics, NAS of Ukraine 2019-03-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/1447 Ukrains’kyi Matematychnyi Zhurnal; Vol. 71 No. 3 (2019); 400-411 Український математичний журнал; Том 71 № 3 (2019); 400-411 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/1447/431 Copyright (c) 2019 Türkmen E. |
| spellingShingle | Türkmen, E. Тюркмен, Є. $\scr{Z^{ \ast}}$ - semilocal modules and the proper class $\scr{RS}$ |
| title | $\scr{Z^{ \ast}}$ - semilocal modules and the proper class $\scr{RS}$ |
| title_alt | $\scr{Z^{ \ast}}$ - напiвлокальнi модулi та власний клас $\scr{RS}$ |
| title_full | $\scr{Z^{ \ast}}$ - semilocal modules and the proper class $\scr{RS}$ |
| title_fullStr | $\scr{Z^{ \ast}}$ - semilocal modules and the proper class $\scr{RS}$ |
| title_full_unstemmed | $\scr{Z^{ \ast}}$ - semilocal modules and the proper class $\scr{RS}$ |
| title_short | $\scr{Z^{ \ast}}$ - semilocal modules and the proper class $\scr{RS}$ |
| title_sort | $\scr{z^{ \ast}}$ - semilocal modules and the proper class $\scr{rs}$ |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/1447 |
| work_keys_str_mv | AT turkmene scrzastsemilocalmodulesandtheproperclassscrrs AT tûrkmenê scrzastsemilocalmodulesandtheproperclassscrrs AT turkmene scrzastnapivlokalʹnimodulitavlasnijklasscrrs AT tûrkmenê scrzastnapivlokalʹnimodulitavlasnijklasscrrs |