Some results on relative dual Baer property

: Let R be a ring. In this article, we introduce and study relative dual Baer property. We characterize R -modules M which are R R -dual Baer, where R is a commutative principal ideal domain. It is shown that over a right noetherian right hereditary ring R , an R -module M is N -dual Baer for all R -modules N if and only if M is an injective R -module. It is also shown that for R -modules M 1 , M 2 , . . . , M n such that M i is M j -projective for all i > j ∈ { 1 , 2 , . . . , n } , an R -module N is (cid:76) ni =1 M i -dual Baer if and only if N is M i -dual Baer for all i ∈ { 1 , 2 , . . . , n } . We prove that an R -module M is dual Baer if and only if S = End R ( M ) is a Baer ring and IM = r M ( l S ( IM )) for every right ideal I of S .


Introduction
Throughout this paper, R will denote an associative ring with identity, and all modules are unitary right R-modules. Let M be an R-module. We will use the notation N M to indicate that N is small in M (i.e., L + N = M for every proper submodule L of M ). By E(M ) and End R (M ), we denote the injective hull of M and the endomorphism ring of M , respectively. By Q, Z, and N we denote the set of rational numbers, integers and natural numbers, respectively. For a prime number p, Z(p ∞ ) denotes the Prüfer p-group.
The concept of Baer rings was first introduced in [6] by Kaplansky. Since then, many authors have studied this kind of rings (see, e.g., [2] and [3]). A ring R is called Baer if the right annihilator of any nonempty subset of R is generated by an idempotent. In 2004, Rizvi and Roman extended the notion of Baer rings to a module theoretic version [10]. According to [10], a module M is called a Baer module if for every left ideal I of End R (M ), ∩ φ∈I Kerφ is a direct summand of M . This notion was recently dualized by Keskin Tütüncü-Tribak in [14]. A module M is said to be dual Baer if for every right ideal I of S = End R (M ), φ∈I Imφ is a direct summand of M . Equivalently, for every nonempty subset A of S, φ∈A Imφ is a direct summand of M (see [14,Theorem 2

.1]).
A module M is said to be Rickart if for any ϕ ∈ End R (M ), Kerϕ is a direct summand of M (see [7]). The notion of dual Rickart modules was studied recently in [8] by Lee-Rizvi-Roman. A module M is said to be dual Rickart if for every ϕ ∈ End R (M ), Imϕ is a direct summand of M . In [8], it was introduced the notion of relative dual Rickart property which was used in the study of direct sums of dual Rickart modules. Let N be an R-module. An R-module M is called N -dual Rickart if for every homomorphism ϕ : M → N , Imϕ is a direct summand of N (see [8]). Similarly, we introduce in this paper the concept of relative dual Baer property. A module M is called N -dual Baer if for every subset A of Hom R (M, N ), We determine the structure of modules M which are R R -dual Baer for a commutative principal ideal domain R (Proposition 2.7). Then we show that for an R-module M , R R is M -dual Baer if and only if M is a semisimple module (Proposition 2.9). It is shown that over a right noetherian right hereditary ring R, an R-module M is N -dual Baer for all R-modules N if and only if M is an injective R-module (Corollary 2.17). We prove that if {M i } I is a family of R-modules, then for each j ∈ I, i∈I M i is M j -dual Baer if and only if M i is M j -dual Baer for all i ∈ I (Corollary 2.24). It is also shown that for R-modules M 1 ,

Main results
Obviously, an R-module M is dual Baer if and only if M is M -dual Baer. (2) If M and N are R-modules such that Hom R (M, N ) = 0, then M is N -dual Baer. It follows that for any couple of different maximal ideals m 1 and m 2 of a commutative noetherian ring R, E(R/m 1 ) is E(R/m 2 )-dual Baer (see [12,Proposition 4.21]).
(3) Let p be a prime number. Note that Z/pZ and Z(p ∞ ) are dual Baer Z-modules. On the other hand, it is clear that Z(p ∞ ) is Z/pZ-dual Baer but Z/pZ is not Z(p ∞ )-dual Baer.
Recall that a module M is said to have the strong summand sum property, denoted briefly by SSSP , if the sum of any family of direct summands of M is a direct summand of M .  Proof. This follows from the definitions of "M is N -d-Rickart" and "M is N -dual Baer".
The next example shows that the assumption "N has the SSSP " is not superfluous in Proposition 2.3.
Example 2.4. Let R be a von Neumann regular ring which is not semisimple (e.g., (iii) Every nonzero ϕ ∈ Hom R (M, N ) is an epimorphism.
Proposition 2.6. Let M and N be modules such that Hom R (M, N ) = 0 (e.g., N is M -generated). Then the following conditions are equivalent: Let π : N → K be the projection map and let i : K → N be the inclusion map. Then i π ϕ ∈ Hom R (M, N ). Assume that i π ϕ = 0. By hypothesis, Imi π ϕ = N . So K = N . Thus K = 0, a contradiction. Therefore i π ϕ = 0. Hence K = 0 and K = N . It follows that N is indecomposable.
The following result describes the structure of R-modules which are R R -dual Baer, where R is a commutative principal ideal domain which is not a field. Proposition 2.7. Let R be a commutative principal ideal domain which is not a field. Then the following conditions are equivalent for an R-module M : (iii) M has no nonzero cyclic torsion-free direct summands; Assume that M has an element x such that xR is a direct summand of M and R R ∼ = xR. Let π : M → xR be the projection map and let f : xR → R R be an isomorphism. Then f π : M → R R is an epimorphism. Let α be a nonzero element of R which is not invertible. Consider the homomorphism g : R R → R R defined by g(r) = αr for all r ∈ R. Then gf π ∈ Hom R (M, R R ) and Imgf π = αR. It is clear that This contradicts our assumption. Hence Hom R (M, R R ) = 0.
In Proposition 2.7, we studied when an R-module M is R R -dual Baer. Next, we investigate when R R is M -dual Baer for an R-module M .
Proposition 2.9. The following conditions are equivalent for an R-module M : Corollary 2.10. The following conditions are equivalent for a ring R: (ii) ⇔ (iii) This follows from Proposition 2.9.
Remark 2.11. If K is a submodule of an R-module M such that K is M -dual Baer, then K is a direct summand of M . In particular, if the R-module M is E(M )-dual Baer, then M is an injective module.
The next example shows that even if a module M is injective, the module M need not be M -dual Baer.
Example 2.12. Let R be a self injective ring which is not semisimple (e.g., R = ∞ n=1 Z/2Z). Then E(R R ) = R R . By [14,Corollary 2.9], the R-module R R is not R R -dual Baer.
Next, we will be concerned with the modules M which are N -dual Baer for all modules N . We begin with the following proposition which provides a class of rings R whose semisimple modules are N -dual Baer for any R-module N . Proof. Let N be an R-module. It is clear that for any ϕ ∈ Hom R (M, N ), Imϕ is semisimple. Let A be a subset of Hom R (M, N ). Then f ∈A Imf is a semisimple submodule of N . Since R is a right noetherian right V-ring, f ∈A Imf is injective by [4,Proposition 1]. Therefore f ∈A Imf is a direct summand of N . So M is N -dual Baer.
The next example shows that the condition "R is a right noetherian ring" in the hypothesis of Proposition 2.13 is not superfluous.
Example 2.14. Let F be a field and let R = n∈N F n such that F n = F for all n ∈ N. Then R is a commutative V-ring which is not noetherian. Note that Soc(R) = ⊕ n∈N F n is an essential proper ideal of R. In particular, Soc(R) is not a direct summand of R. So Soc(R) is not R R -dual Baer.
Following [13], a module M is called noncosingular if for every nonzero module N and every nonzero homomorphism f : M → N , Imf is not a small submodule of N . (ii) ⇒ (i) Let N be an R-module. It is clear that Imϕ is injective for every ϕ ∈ Hom R (M, N ). Since the ring R is right noetherian, f ∈A Imf is injective for every subset A of Hom R (M, N ) by [1,Proposition 18.13]. Therefore f ∈A Imf is a direct summand of N . This proves the proposition.
Recall that a ring R is called right hereditary if each of its right ideals is projective. It is well known that a ring R is right hereditary if and only if every factor module of an injective right R-module is injective (see, for example [16, 39.16]). The next result is a direct consequence of Proposition 2.16. It determines the structure of R-modules M which are N -dual Baer for all R-modules N , where R is a right noetherian right hereditary ring.     (ii) For any direct summand K of M and any submodule N of M , K is N -dual Baer.
From [14, Example 3.1 and Theorem 3.4], it follows that a direct sum of dual Baer modules is not dual Baer, in general. Next, we focus on when a direct sum of N -dual Baer modules is also N -dual Baer for some module N . Proof. Suppose that i∈I M i is N -dual Baer. By Theorem 2.20, M i is N -dual Baer for all i ∈ I. Conversely, assume that M i is N -dual Baer for all i ∈ I. Let {ϕ λ } Λ be a family of homomorphisms in Hom R ( i∈I M i , N ). For each i ∈ I, let µ i : M i → i∈I M i denote the inclusion map. Then for every i ∈ I and every λ ∈ Λ, ϕ λ µ i ∈ Hom R (M i , N ). Since M i is N -dual Baer for every i ∈ I, it follows that Im(ϕ λ µ i ) is a direct summand of N for every (i, λ) ∈ I × Λ. Note that for each λ ∈ Λ, Imϕ λ = i∈I Im(ϕ λ µ i ). As N has the SSSP , λ∈Λ Imϕ λ = λ∈Λ i∈I Im(ϕ λ µ i ) is a direct summand of N . Therefore i∈I M i is N -dual Baer.  Proof. The necessity follows from Theorem 2.20. Conversely, suppose that N is M i -dual Baer for all i ∈ {1, 2, . . . , n}. We will show that N is n i=1 M i -dual Baer. By induction on n and taking into account [9, Proposition 4.33], it is sufficient to prove this for the case n = 2. Assume that N is M i -dual Baer for i = 1, 2 and M 2 is M 1 -projective. Let {φ λ } Λ be a family of homomorphisms in Hom R (N, [9,Proposition 4.32]. As M 1 + λ∈Λ Imφ λ = M 1 ⊕ λ∈Λ π 2 φ λ (N ) is a direct summand of M 1 ⊕ M 2 , there exists a submodule L ≤ λ∈Λ Imφ λ such that M 1 + λ∈Λ Imφ λ = M 1 ⊕L by [9,Lemma 4.47]. Thus λ∈Λ Imφ λ = M 1 ∩ λ∈Λ Imφ λ ⊕L by modularity. It is easily seen that λ∈Λ π 2 φ λ (N ) is a direct summand of M 2 . Let K 2 be a submodule of M 2 such that M 2 = K 2 ⊕ λ∈Λ π 2 φ λ (N ) . Therefore M 1 ⊕M 2 = M 1 ⊕L⊕K 2 . Let π 1 : M 1 ⊕(L⊕K) → M 1 be the projection of M 1 ⊕M 2 on M 1 along L⊕K. Then π 1 φ λ ∈ Hom R (N, M 1 ) for every λ ∈ Λ. Moreover, we have Since each M i (i ∈ I) is dual Baer, each M i (i ∈ I) has the SSSP by Theorem 2.23. Thus λ∈Λ N i,λ is a direct summand of M i for every i ∈ I. So λ∈Λ ϕ λ (M ) is a direct summand of M . Consequently, M is a dual Baer module.
We conclude this paper by showing a new characterization of dual Baer modules.
Let M be an R-module with S = End R (M ). Then for every nonempty subset A of S, we denote l S (A) = {ϕ ∈ S | ϕA = 0} and r M (A) = {m ∈ M | Am = 0}. We also denote l S (N ) = {ϕ ∈ S | ϕ(N ) = 0} for any submodule N of M .
Recall that a ring R is called a Baer ring if for every nonempty subset I ⊆ R, there exists an idempotent e ∈ R such that l S (I) = Re.  (iii) ⇒ (i) Let I be a right ideal of S. Since S is a Baer ring, r M (l S (IM )) is a direct summand of M by Proposition 2.29. But IM = r M (l S (IM )). Then IM is a direct summand of M . By Theorem 2.23, it follows that M is a dual Baer module.