On FI-t-lifting modules
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ORIGINAL ARTICLE
On FI-t-lifting modules Rachid Tribak1 • Yahya Talebi2 • Mehrab Hosseinpour2 • Mona Abdi2 Received: 28 January 2019 / Accepted: 19 June 2020 Ó Sociedad Matemática Mexicana 2020
Abstract In this paper, we introduce the notion of FI-t-lifting modules which is a proper generalization of both the concepts of t-lifting modules and FI-lifting modules. We show that a direct sum of FI-t-lifting modules is not FI-t-lifting, in general. It is also 2 shown that if M is an FI-t-lifting module, then Z ðMÞ is a direct summand of M and 2 Z ðMÞ is a noncosingular FI-lifting module. The last part of the paper is devoted to the study of amply supplemented FI-t-lifting modules. Keywords FI-lifting modules FI-t-lifting modules t-Lifting modules t-Small submodules
Mathematics Subject Classification 16D10 16D80
1 Introduction Throughout this paper, R is an associative ring with unity and M is a unital right Rmodule. The notation N M means that N is a subset of M and N M means that N is a submodule of M. The injective hull of M will be denoted by E(M). By Q, Z, and & Rachid Tribak [email protected] Yahya Talebi [email protected] Mehrab Hosseinpour [email protected] Mona Abdi [email protected] 1
Centre Re´gional des Me´tiers de l’Education et de la Formation (CRMEF-TTH)-Tanger, Avenue My Abdelaziz, Souani, B.P. 3117, Tangier, Morocco
2
Department of Mathematics, Faculty of Mathematical Sciences, University of Mazandaran, Babolsar, Iran
123
R. Tribak et al.
N, we denote the set of rational numbers, integers, and positive integers, respectively. A submodule L of a module M is called small in M (denoted by L M) if, for any X M, M ¼ L þ X implies X ¼ M. Recall that the singular submodule of a module M is the set Z(M) of m 2 M, such that mI ¼ 0 for some essential right ideal I of R. Dually, Talebi and Vanaja introduced in [12] the T submodule ZðMÞ of M defined by ZðMÞ ¼ fU M j M=U EðM=UÞg. The module M is called cosingular (noncosingular) if ZðMÞ ¼ 0 (ZðMÞ ¼ M). Also, for nþ1 n any positive integer n, Z ðMÞ is defined to be ZðZ ðMÞÞ. A submodule K of M is called fully invariant if uðKÞ K for every endomorphism u of M. An R-module M is called lifting if, for every submodule A of M, there exists a direct summand N of M with N A and A=N M=N (see [3]). Let L and N be submodules of a module M. Then, N is called a supplement of L in M if M ¼ L þ N and L \ N N. The module M is said to be amply supplemented if, for any two submodules A and B of M with M ¼ A þ B, B contains a supplement of A in M. It is shown in [9, Proposition 4.8] that a module M is lifting if and only if M is amply supplemented and supplements are direct summands of M. The concept of lifting modules has its roots in the theory of (semi)perfect rings and modules with projective covers (see [3, 27.21 and 27.24]). This concept was first introduced and studied by Takeuchi [11] in 1976, but under the name codirect modules. A number of research papers have been devoted to the study of lifting modules (see [3, Bibliograp
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