Rad-supplements in injective modules
We introduce and study the notion of Rad-s-injective modules (i.e. modules which are Rad-supplements in the irinjective hulls). We compare this notion with another generalization of injective modules. We show that the class of Rad-s-injective modules is closed under finite direct sums. We characteri...
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nasplib_isofts_kiev_ua-123456789-1557372025-02-09T09:51:59Z Rad-supplements in injective modules Buyukasik, E. Tribak, R. We introduce and study the notion of Rad-s-injective modules (i.e. modules which are Rad-supplements in the irinjective hulls). We compare this notion with another generalization of injective modules. We show that the class of Rad-s-injective modules is closed under finite direct sums. We characterize Rad-s-injective modules over several type of rings, including semilocalrings, left hereditary rings and left Harada rings. 2016 Article Rad-supplements in injective modules / E. Buyukasik, R. Tribak // Algebra and Discrete Mathematics. — 2016. — Vol. 22, № 2. — С. 171-183. — Бібліогр.: 14 назв. — англ. 1726-3255 2010 MSC:16D50, 16D99, 16L30, 16L60. https://nasplib.isofts.kiev.ua/handle/123456789/155737 en Algebra and Discrete Mathematics application/pdf Інститут прикладної математики і механіки НАН України |
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We introduce and study the notion of Rad-s-injective modules (i.e. modules which are Rad-supplements in the irinjective hulls). We compare this notion with another generalization of injective modules. We show that the class of Rad-s-injective modules is closed under finite direct sums. We characterize Rad-s-injective modules over several type of rings, including semilocalrings, left hereditary rings and left Harada rings. |
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Buyukasik, E. Tribak, R. |
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Buyukasik, E. Tribak, R. Rad-supplements in injective modules Algebra and Discrete Mathematics |
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Buyukasik, E. Tribak, R. |
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Buyukasik, E. |
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Rad-supplements in injective modules |
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Rad-supplements in injective modules |
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Rad-supplements in injective modules |
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Rad-supplements in injective modules |
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Rad-supplements in injective modules |
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rad-supplements in injective modules |
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Інститут прикладної математики і механіки НАН України |
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Rad-supplements in injective modules / E. Buyukasik, R. Tribak // Algebra and Discrete Mathematics. — 2016. — Vol. 22, № 2. — С. 171-183. — Бібліогр.: 14 назв. — англ. |
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Algebra and Discrete Mathematics |
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AT buyukasike radsupplementsininjectivemodules AT tribakr radsupplementsininjectivemodules |
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2025-11-25T12:55:56Z |
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2025-11-25T12:55:56Z |
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1849767083373494272 |
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Algebra and Discrete Mathematics RESEARCH ARTICLE
Volume 22 (2016). Number 2, pp. 171–183
© Journal “Algebra and Discrete Mathematics”
Rad-supplements in injective modules
Engin Büyükaşık and Rachid Tribak
Communicated by M. Ya. Komarnytskyj
Abstract. We introduce and study the notion of Rad-s-
injective modules (i.e. modules which are Rad-supplements in their
injective hulls). We compare this notion with another generalization
of injective modules. We show that the class of Rad-s-injective
modules is closed under finite direct sums. We characterize Rad-
s-injective modules over several type of rings, including semilocal
rings, left hereditary rings and left Harada rings.
1. Introduction
Throughout this paper all rings are associative with an identity element.
Unless otherwise stated R denotes an arbitrary ring and all modules are
unital left R-modules. Let M be a module. We use N 6 M to denote
that N is a submodule M . The radical of M and the injective hull of M
are denoted by Rad(M) and E(M), respectively. Let N 6 M . We say
that N is small in M (written N ≪ M) if the fact that N + L = M for
some submodule L of M implies L = M . Let N and K be submodules
of M . Then K is said to be a supplement of N in M if N + K = M
and N ∩ K ≪ K. We say that K is a Rad-supplement of N in M
if N + K = M and N ∩ K ⊆ Rad(K). It is clear that the following
implications of conditions on submodules of any module hold:
Direct summands ⇒ supplements ⇒ Rad-supplements.
2010 MSC: 16D50, 16D99, 16L30, 16L60.
Key words and phrases: almost injective modules; Rad-s-injective modules;
injective modules; Rad-supplement submodules.
172 Rad-supplements in injective modules
It is well known that a module M is injective if and only if M is a
direct summand of every module containing M . These modules play an
important role in both module theory and homological algebra. Recently,
a new generalization of injective modules has been studied in [4]. Namely,
a module M is called almost injective if M is a supplement submodule
of every module containing M . So a module is almost injective if and
only if it is a supplement submodule of an injective module containing
it (see [4, Proposition 3.2]). It is of interest to investigate the analogue
of this notion by replacing “supplement”with “Rad-supplement”. We call
a module M Rad-s-injective, if M is a Rad-supplement in every module
containing M .
Proposition 1.1. The following are equivalent for a module M :
(i) M is a Rad-s-injective module;
(ii) There exists an injective module E containing M such that M is
a Rad-supplement in E;
(iii) M is a Rad-supplement in its injective hull E(M).
Proof. This follows from [2, Proposition 4.3(i),(iii)].
Clearly every almost injective module is Rad-s-injective. We begin
with an example which shows a Rad-s-injective module which is not almost
injective. Then we investigate the structure of Rad-s-injective modules
over some type of rings. We show that the Rad-s-injective property always
transfer from a module to each of its direct summands. It is shown that the
class of Rad-s-injective modules is closed under finite direct sums, but it is
not closed under factor modules. We prove that over a left hereditary ring
R, an R-module M is Rad-s-injective if and only if M/Rad(M) is injective.
Another characterization of Rad-s-injective modules over semilocal rings
is provided.
2. Results
Let M be an R-module with Rad(M) = M . Note that M is a Rad-
supplement of E(M) in E(M). So, M is a Rad-s-injective module.
It is obvious that every almost injective (in particular, every injective)
module is Rad-s-injective. The following example shows that there are
Rad-s-injective modules which are not almost injective.
Example 2.1. Let M be a module such that Rad(M) = M and M ≪
E(M). Then M is a Rad-s-injective module, but is not an almost injective
E. Büyükaşık, R. Tribak 173
module. To construct a module satisfying these conditions, let F be a
field and let R be the ring of polynomials in countably many commuting
variables x1, x2, . . ., over F subject to the relations x2
1 = 0 and x2
n = xn−1
for n > 2. This ring appears in [13] in another context. The ring R is local
with maximal ideal J generated by the xi. Moreover, we have J2 = J .
Therefore Rad(RJ) =R J ≪R R. It follows that the R-module RJ is
Rad-s-injective, while RJ is not almost injective.
In the following result, we provide a condition under which a Rad-s-
injective module is almost injective.
Proposition 2.2. Let M be an R-module with Rad(M) ≪ M . Then M
is Rad-s-injective if and only if M is an almost injective module.
Proof. Assume that M is Rad-s-injective. Then there exists a submodule
K of E(M) such that K + M = E(M) and K ∩ M ⊆ Rad(M). Since
Rad(M) ≪ M , we have K∩M ≪ M . It follows that M is a supplement of
K in E(M). Therefore M is almost injective. The converse is immediate.
Corollary 2.3. Let M be a module over a left perfect ring. Then M is
Rad-s-injective if and only if M is almost injective.
Proof. This follows from Proposition 2.2 and [1, Remark 28.5(3)].
Let R be a commutative ring. Then an element a ∈ R is said to be a
zero-divisor in case there is an element b 6= 0 in R with ba = 0.
The next result shows that [4, Proposition 3.16] still holds if it is
assumed that R is a commutative ring.
Proposition 2.4. Let R be a commutative ring. Assume that there ex-
ists an index set I such that the R-module R(I) is Rad-s-injective and
Rad(R(I)) ≪ R(I). Then, for any x ∈ R, x is invertible if and only if x
is not a zero-divisor in R.
Proof. Let E be the injective hull of R(I). By Proposition 2.2, there exists
a submodule M of E such that R(I) is a supplement of M in E. Let r be
an element of R which is not a right zero-divisor. Then rM + rR(I) = rE.
By [12, Proposition 2.6], we have rE = E. Thus M + (rR)(I) = E. By
the minimality of R(I), we have (rR)(I) = R(I). So rR = R. This implies
that r is invertible. The converse is obvious.
174 Rad-supplements in injective modules
Corollary 2.5. Let R be a commutative domain. Then the following
conditions are equivalent:
(i) RR is a Rad-s-injective module;
(ii) RR is an almost injective module;
(iii) R is a division ring.
Proof. This follows from Propositions 2.2 and 2.4.
Next, we investigate the structure of Rad-s-injective modules over
some type of rings. We first state the following lemma.
Lemma 2.6 (see [2, Corollary 4.2]). Let N be a submodule of an R-module
M . If N is a Rad-supplement in M , then Rad(N) = N ∩ Rad(M).
Recall that a ring R is called left small if the R-module RR is small
in its injective hull E(RR). By [9, Proposition 2.4], a ring R is left small
if and only if Rad(E) = E for any injective R-module E.
Proposition 2.7. The following statements are equivalent for a ring R:
(i) R is a left small ring;
(ii) For any R-module M , M is Rad-s-injective if and only if
Rad(M) = M .
Proof. (i) ⇒ (ii) Let M be a Rad-s-injective R-module. Then M is a Rad-
supplement in its injective hull E(M). Since R is left small, Rad(E(M)) =
E(M) by [9, Proposition 2.4]. So Rad(M) = M ∩ Rad(E(M)) = M ∩
E(M) = M by Lemma 2.6. This is the desired conclusion.
(ii) ⇒ (i) This follows from [9, Proposition 2.4] and the fact that every
injective module is Rad-s-injective.
Combining Proposition 2.7 and [8, Theorem 2], we obtain the following
corollaries.
Corollary 2.8. Let R be a commutative domain which is not a field. If M
is an R-module, then M is Rad-s-injective if and only if Rad(M) = M .
Corollary 2.9. Let M be a module over a Dedekind domain R. Then M
is Rad-s-injective if and only if M is injective.
A ring R is called a left max (or left Bass) ring if Rad(M) 6= M for
every nonzero left R-module M . It is well known that a ring R is left max
if and only if Rad(M) ≪ M for every nonzero left R-module M . From
Proposition 2.2, it follows that for a module M over a left max ring, M
is Rad-s-injective if and only if M is almost injective.
E. Büyükaşık, R. Tribak 175
Proposition 2.10. The following conditions are equivalent for a ring R:
(i) R is a left max ring;
(ii) Rad(M) 6= M for any Rad-s-injective R-module M .
Proof. This follows from the fact that every R-module M with Rad(M) =
M , is Rad-s-injective.
A ring R is called a left Harada ring if R is left artinian and every
nonsmall R-module contains a nonzero injective direct summand (see, for
example, [5, 28.10]). Note that quasi-Frobenius rings and artinian serial
rings are left Harada rings. Recall that a module M is called small if
M ≪ L for some R-module L. By [9, Proposition 2.1], a module M is a
small module if and only if M ≪ E(M).
Proposition 2.11. Let R be a left Harada ring. Then every Rad-s-
injective R-module is injective.
Proof. Note that R is a perfect ring by [5, 28.10]. Let M be a Rad-s-
injective R-module. Therefore M is almost injective by Corollary 2.3.
Using again [5, 28.10], M = N ⊕K such that N is an injective module
and K is a small module. But K is almost injective by [4, Corollary 3.3].
Then K = 0 and M is an injective module. This completes the proof.
Corollary 2.12. The following conditions are equivalent for a ring R:
(i) R is a quasi-Frobenius ring;
(ii) Every Rad-s-injective R-module M is projective.
Proof. This follows from Proposition 2.11 and [1, Theorem 31.9].
It is of natural interest to investigate if or not the class of Rad-s-
injective modules is closed under submodules, direct summands, direct
sums and factor modules. The next example shows that the Rad-s-injective
property is not inherited by submodules, in general.
Example 2.13. Consider the Z-module M = Q and the submodule
N = Z of M . Note that M is an injective module, N ≪ M and Rad(N) =
0. Then M is Rad-s-injective, but N is not a Rad-s-injective module.
Proposition 2.14. Every Rad-supplement submodule of a Rad-s-injective
module is Rad-s-injective.
Proof. Let M be a Rad-s-injective module and let K be a Rad-supplement
in M . Since M is a Rad-supplement in E(M), K is a Rad-supplement in
E(M) by [2, Proposition 4.3(iii)]. The result follows from Proposition 1.1.
176 Rad-supplements in injective modules
From the last result, it follows that a direct summand of a Rad-s-
injective module is Rad-s-injective. The next example exhibits that the
direct sum of Rad-s-injective modules does not always inherit the property.
Example 2.15. Let F be a field and let the commutative ring R be the
direct product
∏
∞
n=1 Fn, where Fn = F for each n ∈ N. Then R is a von
Neumann regular ring. It is easy to see that the ideal I =
⊕
∞
n=1 Fn is not
a direct summand of RR. Since Rad(RI) = 0, the R-module RI is not a
Rad-supplement submodule of RR. Therefore RI is not a Rad-s-injective
module. On the other hand, it is clear that the R-module RFn is injective
for each n ∈ N.
Next, in order to show that a finite direct sum of Rad-s-injective
modules is also Rad-s-injective, we need the following lemma.
Lemma 2.16 (see [6, Lemma 2.13]). Let X, Y and Z be submodules of
a module M such that M = X + Y + Z. If X is a Rad-supplement of
Y + Z in M and Y is a Rad-supplement of X + Z in M , then X + Y is
a Rad-supplement of Z in M .
Proof. By assumption, we have X∩(Y +Z) ⊆ Rad(X) and Y ∩(X+Z) ⊆
Rad(Y ). Since Z ∩ (X + Y ) ⊆ [X ∩ (Y + Z)] + [Y ∩ (X + Z)], we have
Z ∩ (X + Y ) ⊆ Rad(X) + Rad(Y ). Thus Z ∩ (X + Y ) ⊆ Rad(X + Y ).
This proves the lemma.
Proposition 2.17. A finite direct sum M1 ⊕ · · · ⊕Mn is Rad-s-injective
if and only if Mi is Rad-s-injective for each i = 1, . . . , n.
Proof. Without loss of generality we can assume that n = 2. Let M =
M1 ⊕ M2. Then E(M) = E(M1) ⊕ E(M2). Assume that M1 and M2
are Rad-s-injective. Therefore there exist submodules N1 6 E(M1) and
N2 6 E(M2) such that Mi is a Rad-supplement of Ni in E(Mi) (i =
1, 2). Note that M1 ∩ (N1 + N2 + M2) = M1 ∩ N1 ⊆ Rad(M1) and
M2 ∩ (N1 + M1 + N2) = M2 ∩ N2 ⊆ Rad(M2). Then M1 is a Rad-
supplement of N1 +N2 +M2 in E(M) and M2 is a Rad-supplement of
N1+M1+N2 in E(M). Therefore M1+M2 is a Rad-supplement ofN1+N2
in E(M) by Lemma 2.16. Consequently, M1 ⊕ M2 is a Rad-s-injective
module. The converse follows from Proposition 2.14.
The next result will be of interest.
Proposition 2.18. The following statements are equivalent for an R-
module M :
E. Büyükaşık, R. Tribak 177
(i) M is Rad-s-injective;
(ii) M/Rad(M) is a direct summand of E(M)/Rad(M).
Proof. (i) ⇒ (ii) By assumption, there exists a submodule N of E(M)
such that M +N = E(M) and M ∩N ⊆ Rad(M). Therefore,
[M/Rad(M)] + [(N + Rad(M))/Rad(M)] = E(M)/Rad(M)
and
M ∩ (N + Rad(M)) = Rad(M) + (M ∩N) = Rad(M).
Thus,
[M/Rad(M)] ⊕ [(N + Rad(M))/Rad(M)] = E(M)/Rad(M).
This is our claim.
(ii) ⇒ (i) Let N be a submodule of E(M) such that Rad(M) ⊆ N
and (M/Rad(M))⊕(N/Rad(M)) = E(M)/Rad(M). Therefore M+N =
E(M) and M ∩N ⊆ Rad(M), i.e. M is a Rad-supplement of N in E(M).
This completes the proof.
The next result is a direct consequence of Proposition 2.18.
Corollary 2.19. (i) Let M be an R-module with Rad(M) = 0. Then M
is Rad-s-injective if and only if M is an injective module.
(ii) A semisimple R-module M is Rad-s-injective if and only if M is
an injective module.
(iii) Let M be an R-module such that M/Rad(M) is injective. Then
M is Rad-s-injective.
Recall that a ring R is called a left V -ring if every simple left R-module
is injective or, equivalently, Rad(M) = 0 for every left R-module M . The
next result is a direct consequence of Corollary 2.19(i).
Corollary 2.20. Let R be a left V -ring. For any R-module M , M is
Rad-s-injective if and only if M is injective.
Proposition 2.21. The following are equivalent for a ring R:
(i) Every R-module is Rad-s-injective;
(ii) Every factor module of RR is Rad-s-injective;
(iii) The ring R is semisimple.
178 Rad-supplements in injective modules
Proof. (iii) ⇒ (i) ⇒ (ii) are immediate.
(ii) ⇒ (iii) By hypothesis, every simple R-module is Rad-s-injective.
Therefore every simple R-module is injective (Corollary 2.19(ii)). This
implies that R is a left V -ring. So Rad(M) = 0 for any R-module M . By
(ii) and Corollary 2.19(i), it follows that every cyclic R-module is injective.
Hence R is semisimple by [11, Theorem on p. 649].
The next example shows that the class of Rad-s-injective modules is
not always closed under factor modules.
Example 2.22. Let R be a ring such that RR is a Rad-s-injective module,
but R is not semisimple. For example, we can take a perfect local ring R
which is not semisimple by [4, Example 2.3]). Then the module RR has
a factor module which is not Rad-s-injective by Proposition 2.21. Note
that some examples of this type of rings are cited in [4, Examples 2.7].
Proposition 2.23. Let L be a submodule of a module M such that
L ⊆ Rad(M). Assume that M/L is Rad-s-injective. Then M is Rad-s-
injective.
Proof. By assumption, there exists a submodule N of E(M) such that
L ⊆ N and M/L is a Rad-supplement of N/L in E(M)/L. Then
(M ∩N)/L = (M/L) ∩ (N/L) ⊆ Rad(M/L) = Rad(M)/L.
Therefore M ∩N ⊆ Rad(M). Since M +N = E(M), it follows that M
is a Rad-s-injective module.
The following example illustrates that the condition “L ⊆ Rad(M)”in
the hypothesis of Proposition 2.23 is not superfluous.
Example 2.24. It is well known that the Z-module Q ∼= Z(I)/L for some
index set I and a submodule L of the Z-module Z(I). Since Q is injective,
Z(I)/L is a Rad-s-injective Z-module. On the other hand, Z(I) is not
Rad-s-injective, since otherwise the Z-module ZZ will be Rad-s-injective
by Proposition 2.14. Then ZZ will be injective as Rad(ZZ) = 0 (see
Corollary 2.19(i)), a contradiction.
Proposition 2.25. Let M1 and M2 be two submodules of a module M
such that M = M1 + M2 and M1 ∩ M2 ⊆ Rad(M). Suppose that every
factor module of Mi is Rad-s-injective for each i = 1, 2. Then M is a
Rad-s-injective module.
E. Büyükaşık, R. Tribak 179
Proof. Note that M/M1 ∩ M2 = (M1/M1 ∩ M2) ⊕ (M2/M1 ∩ M2). By
hypothesis, M1/M1 ∩ M2 and M2/M1 ∩ M2 are Rad-s-injective. Thus
M/M1 ∩ M2 is Rad-s-injective by Proposition 2.17. The result follows
from Proposition 2.23.
Recall that a ring R is said to be left hereditary if every left ideal of
R is a projective R-module. The next result characterizes Rad-s-injective
modules over left hereditary rings.
Theorem 2.26. Let R be a left hereditary ring. Then the following
conditions are equivalent for an R-module M :
(i) M is a Rad-s-injective R-module;
(ii) Every factor module of M is a Rad-s-injective R-module;
(iii) M/Rad(M) is an injective R-module.
Proof. (i) ⇒ (ii) Let N be a submodule of M . By hypothesis, there
exists a submodule L of E(M) such that L+M = E(M) and L ∩M ⊆
Rad(M). Therefore ((L + N)/N) + (M/N) = E(M)/N . Moreover, by
[1, Proposition 9.14], we have
((L+N)/N) ∩ (M/N)
= [N + (L ∩M)]/N ⊆ [N + Rad(M)]/N ⊆ Rad(M/N).
This implies that M/N is a Rad-supplement of (L+N)/N in E(M)/N .
Since R is left hereditary, E(M)/N is an injective R-module by [14, 39.16].
Thus M/N is a Rad-s-injective R-module by Proposition 1.1.
(ii) ⇒ (iii) This follows from Corollary 2.19(i) and [1, Proposition 9.15].
(iii) ⇒ (i) This follows from Corollary 2.19(iii).
Matlis showed that a ring R is left hereditary if and only if the sum
of two injective submodules of any left R-module is injective (see, for
example, [10, Exercise 10 on p. 114] or [12, Exercise 2.11]). Next, we
examine the Rad-s-injective analogue of this characterization.
Proposition 2.27. If R is a left hereditary ring, then the sum of two
Rad-s-injective submodules of any R-module is Rad-s-injective.
Proof. Assume that R is a left hereditary ring. Let M = M1 +M2 such
that M1 and M2 are Rad-s-injective. Then
M/Rad(M) = [(M1 + Rad(M))/Rad(M)] + [(M2 + Rad(M))/Rad(M)].
180 Rad-supplements in injective modules
Let i ∈ {1, 2}. Note that
(Mi + Rad(M))/Rad(M) ∼= Mi/(Mi ∩ Rad(M)).
Since Rad(Mi) ⊆ Mi ∩ Rad(M), Mi/(Mi ∩ Rad(M)) is a factor module
of Mi/Rad(Mi). Since Mi is Rad-s-injective, Mi/Rad(Mi) is injective by
Theorem 2.26. Therefore Mi/(Mi ∩ Rad(M)) is also injective as R is left
hereditary. It follows that M/Rad(M) is an injective module. Applying
again Theorem 2.26, we conclude that M is a Rad-s-injective module.
It is shown in [10, Theorem (Bass, Papp) 3.46] that the left noetherian
rings are exactly the rings over which every direct sum of injective modules
is injective (see also [1, Proposition 18.13]). One may ask whether this
is still true if we replace “injective”with “Rad-s-injective”in this result.
Next, we investigate this question.
Proposition 2.28. If R is a left noetherian ring, then every direct sum
of Rad-s-injective R-modules is Rad-s-injective.
Proof. Assume that R is a left noetherian ring. Let (Mi)i∈I be an indexed
set of submodules of an R-module M such that M =
⊕
i∈I Mi and Mi is
a Rad-s-injective module for each i ∈ I. Then E(M) =
⊕
i∈I E(Mi) and
Rad(M) =
⊕
i∈I Rad(Mi) by [1, Propositions 9.19 and 18.13]. So there
exists an isomorphism ψ : E(M)/Rad(M) →
⊕
i∈I(E(Mi)/Rad(Mi))
such that ψ(M/Rad(M)) =
⊕
i∈I(Mi/Rad(Mi)). By Proposition 2.18,
Mi/Rad(Mi) is a direct summand of E(Mi)/Rad(Mi) for each i ∈ I. It
follows that M/Rad(M) is a direct summand of E(M)/Rad(M). Again
by Proposition 2.18, M is a Rad-s-injective module. This proves the
proposition.
The following example shows that both the converses of Proposi-
tions 2.27 and 2.28 are not true, in general.
Example 2.29. Let R be a commutative domain which is not noetherian
(e.g., we can take the polynomial ring R = F [X1, . . . , Xn, . . .] in infinitely
many indeterminates over a field F or we can find other examples in [7,
Examples 1.13 and 1.14 on p. 8]). Thus R is not a Dedekind domain. So the
ring R is not hereditary. By [8, Theorem 2], the R-module RR is small in
its injective hull E(RR). Therefore the class of Rad-s-injective R-modules
is exactly the class of modules M with Rad(M) = M (see Proposition 2.7).
It follows easily that the sum of two Rad-s-injective submodules of any
R-module is Rad-s-injective. Also, note that the factor modules of any
E. Büyükaşık, R. Tribak 181
Rad-s-injective R-module are Rad-s-injective and every direct sum of
Rad-s-injective modules is Rad-s-injective (see [1, Proposition 9.19]).
Remark 2.30. Example 2.29 illustrates that if R is a ring such that
every factor module of a Rad-s-injective module is Rad-s-injective, then
the ring R need not be left hereditary. This should be contrasted with
[10, Theorem 3.22].
Recall that a ring R is called semilocal if R/Rad(R) is semisimple.
Recall that if M is a module over a semilocal ring R with J = Rad(R),
then Rad(M) = JM and the module M/JM is semisimple. In the next
proposition, we provide a characterization of Rad-s-injective modules over
semilocal rings.
Proposition 2.31. Let R be a semilocal ring with Jacobson radical J
and let M be an R-module. Then M is Rad-s-injective if and only if
JM = M ∩ JE(M).
Proof. The necessity follows from Lemma 2.6 and [1, Corollary 15.18].
Conversely, note that the module E(M)/Rad(E(M)) is semisimple as
R is a semilocal ring. Therefore (M + Rad(E(M)))/Rad(E(M)) is a
direct summand of E(M)/Rad(E(M)). Let K be a submodule of E(M)
such that Rad(E(M)) ⊆ K and [(M + Rad(E(M)))/Rad(E(M))] ⊕
[K/Rad(E(M))] = E(M)/Rad(E(M)). So M+K = E(M) and M∩K ⊆
Rad(E(M)). Moreover, we have M ∩ K ⊆ Rad(M) since Rad(M) =
M ∩ Rad(E(M)). Thus M is a Rad-supplement of K in E(M). Hence M
is Rad-s-injective.
Recall that if N is a submodule of a module M , then N is said to be
a pure submodule of M if IN = N ∩ IM for each right ideal I of R (see
[1, p. 232]).
Corollary 2.32. Let R be a semilocal ring and E be an injective R-
module. Then every pure submodule of E is a Rad-s-injective module.
Proof. Let J be the Jacobson radical of R and let M be a pure submodule
of E. Then JM = M ∩ JE. This gives JM = M ∩ JE(M). Thus M is a
Rad-s-injective module by Proposition 2.31.
A module M is said to be w-local if it has a unique maximal sub-
module (see [3]). Clearly, every local module is w-local. The next result
characterizes w-local Rad-s-injective modules.
182 Rad-supplements in injective modules
Proposition 2.33. Let M be a w-local module with maximal submod-
ule K. Then the following conditions are equivalent:
(i) M is a Rad-s-injective module;
(ii) M 6⊆ Rad(E(M)).
Proof. Note that if X is a submodule of E(M) such that M +X = E(M),
then M/(X ∩M) ∼= E(M)/X. Using this fact, we see that
“M is a Rad-s-injective module.”
m
“There is a submodule N of E(M) such that M + N = E(M) and
M ∩N ⊆ K.”
m
“E(M) has a maximal submodule L such that M + L = E(M).”
m
“E(M) has a maximal submodule L such that M 6⊆ L.”
m
“M 6⊆ Rad(E(M)).”
Corollary 2.34. Let M = ⊕n
i=1Mi such that each Mi is a w-local sub-
module of M . Then M is Rad-s-injective if and only if Mi 6⊆ Rad(E(M))
for each i = 1, . . . , n.
Proof. This follows from Lemma 2.6 and Propositions 2.17 and 2.33.
The next corollary is a direct consequence of Proposition 2.33.
Corollary 2.35. Let R be a left small ring (e.g. R is a Dedekind domain
which is not a field). Then R has no w-local Rad-s-injective modules.
References
[1] F.W. Anderson and K.R. Fuller, Rings and Categories of Modules, Graduate texts
in mathematics 13, Springer-Verlag, New York, 1992.
[2] E. Büyükaşık, E. Mermut and S. Özdemir, Rad-supplemented modules, Rend.
Semin. Mat. Univ. Padova, 124(2010), 157-177.
[3] E. Büyükaşık and C. Lomp, On a recent generalization of semiperfect rings, Bull.
Aust. Math. Soc., 78(2)(2008), 317-325.
[4] J. Clark, D. K. Tütüncü and R. Tribak, Supplement submodules of injective
modules, Comm. Algebra, 39(11)(2011), 4390-4402.
[5] J. Clark, C. Lomp, N. Vanaja and R. Wisbauer, Lifting Modules. Supplements and
Projectivity in Module Theory, Frontiers in Mathematics, Birkhäuser, Basel, 2006.
[6] Ş. Ecevit, M. T. Koşan and R. Tribak, Rad-⊕-supplemented modules and cofinitely
Rad-⊕-supplemented modules, Algebra Colloq., 19(4)(2012), 637-648.
E. Büyükaşık, R. Tribak 183
[7] L. Fuchs and L. Salce, Modules over non-Noetherian domains, Mathematical
surveys and monographs 84, American Mathematical Society, 2001
[8] M. Harada, On small submodules in the total quotient ring of a commutative ring,
Rev. Un. Mat. Argentina, 28(1977), 99-102.
[9] M. Harada, A note on hollow modules, Rev. Un. Mat. Argentina, 28(1978), 186-194.
[10] T.Y. Lam, Lectures on Modules and Rings, Graduate Texts in Mathematics 189,
Springer-Verlag, New York, 1998.
[11] B.L. Osofsky, Rings all of whose finitely generated modules are injective, Pacific J.
Math., 14(1964), 645-650.
[12] D.W. Sharpe and P. Vámos, Injective Modules, Cambridge University Press, 1972.
[13] H.H. Storrer, On Goldman’s primary decomposition, Lecture Notes in Math.,
246(1972), 617-661.
[14] R. Wisbauer, Foundations of Module and Ring Theory, Gordon and Breach,
Philadelphia, 1991.
Contact information
Engin Büyükaşık İzmir Institute of Technology,
Department of Mathematics,
35430, Urla, İzmir-TURKEY
E-Mail(s): enginbuyukasik@iyte.edu.tr
Rachid Tribak Centre Régional des Métiers de l’Education
et de la Formation (CRMEF)-Tanger,
Avenue My Abdelaziz, Souani, BP : 3117,
Tanger, Morocco
E-Mail(s): tribak12@yahoo.com
Received by the editors: 10.12.2015
and in final form 08.03.2016.
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