PRIMARY SUBMODULES OVER A MULTIPLICATIVELY CLOSED SUBSET OF A COMMUTATIVE RING

Authors

  • Nahid Ilaghi Payame Noor University
  • Mansooreh Maani-Shirazi Payame Noor University
  • Shamsolmolouk Khoshdel Payame Noor University

DOI:

https://doi.org/10.30495/jme.v0i0.1356

Keywords:

Multiplicatively closed subset, Multiplication module, Primary module, S-primary module

Abstract

In this paper, we introduce the concept of primary submodules overS which is a generalization of the concept of S-prime submodules. Suppose S isa multiplicatively closed subset of a commutative ring R and let M be a unitalR-module. A proper submodule Q of M with (Q :R M) \ S = ; is called primaryover S if there is an s 2 S such that, for all a 2 R, m 2 M, am 2 Q implies thatsm 2 Q or san 2 (Q :R M), for some positive integer n. We get some new resultson primary submodules over S. Furtheremore, we compare the concept of primarysubmodules over S with primary ones. In particular, we show that a submoduleQ is primary over S if and only if (Q :M s) is primary, for some s 2 S.

Author Biographies

Nahid Ilaghi, Payame Noor University

Department of Mathematics

Ph.D. Student

Mansooreh Maani-Shirazi, Payame Noor University

Department of Mathematics

Assistant of Professor

Shamsolmolouk Khoshdel, Payame Noor University

Department of Mathematics

Assistant of Professor

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Published

2020-07-22

Issue

Section

Vol. 15, No. 2, (2021)