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Proc. Indian Acad. Sci. (Math. Sci.) (2018) 128:20 https://doi.org/10.1007/s12044-018-0403-6 On a generalization of semisimple modules NAZIM AGAYEV1,2,∗, CESIM˙ ÇELIK˙ 1,2 and TAHIRE˙ ÖZEN1,2 1Computer Engineering Department, Fatih Sultan Mehmet Vakf University, Istanbul, Turkey 2Department of Mathematics, Abant Izzet˙ Baysal University, Gölköy Kampüsü, Bolu, Turkey *Corresponding author. E-mail: [email protected]; [email protected]; [email protected] MS received 28 January 2016; revised 14 May 2016; accepted 25 August 2016; published online 24 April 2018 Abstract. Let R be a ring with identity. A module MR is called an r-semisimple module if for any right ideal I of R, MI is a direct summand of MR which is a generalization of semisimple and second modules. We investigate when an r-semisimple ring is semisimple and prove that a ring R with the number of nonzero proper ideals ≤4 and J(R) = 0isr-semisimple. Moreover, we prove that R is an r-semisimple ring if and only if it is a direct sum of simple rings and we investigate the structure of module whenever R is an r-semisimple ring. Keywords. Semisimple modules; second modules; injective modules. 1991 Mathematics Subject Classification. 16U80. 1. Introduction Throughout, we consider associative rings with identity and all modules are unitary. For a module M over a ring R, we write MR to indicate that M is a right R-module. The Socle and the Jacobson radical of M are denoted by Soc(M) and J(M), respectively. The module M is called semisimple if Soc(M) = M. In section 2, we define and characterize r-semisimple modules and rings which are generalizations of semisimple and second modules and rings. A module MR is called a semisimple module if every submodule is a direct summand of MR. A right R-module M is second if M = 0 and annR(M) = annR(M/N) for every proper submodule N of M (see [9]). In [2], it is proved that a module MR is a second module if for every ideal I of R either MI = 0orMI = M. A module MR is called an r-semisimple module if for any right ideal I of R, MI is a direct summand of MR (or equivalently for every ideal I , MI is a direct summand of MR). It is clear that every semisimple and second modules are r-semisimple but the converse may not be true. If RR is r-semisimple, then it is semiprimitive, that is, J(R) = 0. Every direct summand and direct sum of r-semisimple modules are also r-semisimple. We investigate when an r-semisimple ring is semisimple and simple. Moreover, any module MR need not be r-semisimple although every proper © Indian Academy of Sciences 1 20 Page 2 of 10 Proc. Indian Acad. Sci. (Math. Sci.) (2018) 128:20 submodule is r-semisimple. We show that a ring R with the number of the proper right ideals ≤4 and J(R) = 0isr-semisimple. In section 3, we investigate the structure of ring and modules whenever R is an r- semisimple ring. We obtain that R is an r-semisimple ring if and only if it is a direct sum of simple rings and we prove that every injective module MR is r-semisimple and if RR is also self-injective and noetherian, then RR is semisimple . If R is an r-semisimple and right Kasch ring, then R is semisimple. For any unexplained terminology, we refer the reader to [1], [7] and [2]. 2. Preliminaries In this section, we study some elementary properties of r-semisimple modules. DEFINITION 2.1 Let M be a right R-module. M is called an r-semisimple module if for any right ideal I of R, MI is a direct summand of MR. The ring R is said to be an r-semisimple ring if RR is r-semisimple. It is clear that M is an r-semisimple right R module if and only if for any ideal I , MI is a direct summand of MR. Let us give some examples of r-semisimple rings and modules: (1) Clearly, every simple ring and semisimple ring (module) is an r-semisimple ring (module). Moreover, Zp∞ is an r-semisimple module, but not semisimple. (2) Second modules (see [2]) are r-semisimple, but the converse is not true. For example, M = Z2 ⊕ Z3 is an r-semisimple Z-module but is not second. (3) If R is both commutative and r-semisimple ring, then it is semisimple. (4) Let M be a torsion-free and r-semisimple module. Then M is second. Suppose, M is not second, then there exists a proper nonzero right submodule N of M and right ideal I such that MI ⊕ N = M. Then it implies NI = 0, which is a contradiction. (5) If M is a non-singular r-semisimple module on a uniform ring R, then M is second. If M is not second, then there exists a proper nonzero right submodule N and right ideal I such that MI ⊕ N = M. Then NI = 0 and so for any 0 = n ∈ N, I ≤ annR(n), but this contradicts the fact that M is non-singular. (6) Let R be a domain and r-semisimple ring. Then R is second and hence, it is simple. Moreover, xaR = xR for every minimal left ideal xR and any nonzero element a. Thus, if R is not a division ring, then Soc(RR) = 0. (7) Let R be a uniform r-semisimple ring. Then R is simple. Lemma 2.2. Let R be an r-semisimple ring. Then R is a semiprimitive ring. Proof. R is an r-semisimple ring, so J(R) is a direct summand of R. Since J(R) is a small submodule of R,wehaveJ(R) = 0. So, R is semiprimitive. Example 2.3. The inverse of Lemma 2.2 is not true. Let R = Z. Then J(Z) = 0, but Z is not an r-semisimple ring. Proc. Indian Acad. Sci. (Math. Sci.) (2018) 128:20 Page 3 of 10 20 COROLLARY 2.4 If R is an artinian and r-semisimple ring, then R is semisimple. Proof. It follows from Proposition 15.16 in [1]. Lemma 2.5. Let R be an r-semisimple ring and has at least three maximal right ideals. If R has a maximal and minimal right ideal I , then RI = R. Proof. Let I, I1, I2 be three different maximal right ideals of R. Assume that RI = R. Then RI = I , hence I is a two sided ideal of R. Since I is both maximal and minimal ideal, I ∩ I1 = I ∩ I2 = 0. Therefore I1 I = I2 I = 0 and I ⊆ RI = (I1 + I2)I ⊆ I1 I + I2 I = 0. But this is a contradiction. So RI = R. COROLLARY 2.6 Let R be an r-semisimple ring and every proper right ideal be maximal. If R has at least three right ideals, then R is simple. Lemma 2.7. Let R be r-semisimple and for every proper right ideal I, RI = R. Then R is semisimple. Proof. Let I be any proper right ideal of R. Then I cannot be an essential ideal of R.By Lemma 1 in [8], either I is a direct summand or there exists a proper essential right ideal L of R containing I which implies that RL = R. Then I is a direct summand of R, whence R is semisimple. Lemma 2.8. Let R be r-semisimple and u · dim(RR)<∞. Then Soc(RR) is a direct summand of R. Proof. We can assume u ·dim(RR) ≤ 2. Then Soc(RR) = I1 ⊕ I2 where all Ii are minimal right ideals for i = 1, 2. Since J(R) = 0, there exists a maximal right ideal M1 of R such that M1 ∩ I1 = 0 and so M1 ⊕ I1 = R. Therefore Soc(M1) is simple and J(M1) = 0. Then there exists a maximal right submodule M2 of M1 such that Soc(M1) ∩ M2 = 0 and Soc(M1) ⊕ M2 = M1. Thus R = I1 ⊕ Soc(M1) ⊕ M2 = Soc(RR) ⊕ M2. Lemma 2.9. Infinitely many direct product of r-semisimple modules is r-semisimple if every ideal of R is a finitely generated left ideal. Moreover, a direct sum of r-semisimple modules is r-semisimple too. Proof. See [1], page 174. Example 2.10. Let R = Z2×Z3×···×Zp×···; p is prime (ring with an identity element). Then R is not semisimple. Since every commutative r-semisimple ring is semisimple, R can not be an r-semisimple ring. But, clearly, R is an r-semisimple Z-module. Moreover, if R = Z2 ⊕ Z3 ⊕···⊕Zp ⊕···; p is prime (which is a ring without identity), then it is an r-semisimple Z-module, too. 20 Page 4 of 10 Proc. Indian Acad. Sci. (Math. Sci.) (2018) 128:20 Example 2.11. Let M = ZZ. Then E(M) = Q. It is clear that E(M) is an r-semisimple Z- module. But M is notr-semisimple. This example shows that, submodule of ar-semisimple module need not be r-semisimple. Moreover, it is not extension closed which follows from the exact sequence 0 → Z2 → Z4 → Z2 → 0. The following example shows that there exists a simple nonreduced r-semisimple ring with zero divisors and with only three both maximal and minimal right ideals, where RI = R for all nonzero right ideals I . Example 2.12. Let R = M2(Z2). R has zero divisors. The proper right ideals of R are ab 00 I = |a, b ∈ Z , I = |a, b ∈ Z 1 00 2 2 ab 2 ab I = |a, b ∈ Z .
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