# ASSOCIATED PRIME SUBMODULES OF A MULTIPLICATION MODULE

• Lee, Sang Cheol (Department of Mathematics Education, and Institute of Pure and Applied Mathematics, Chonbuk National University) ;
• Song, Yeong Moo (Department of Mathematics Education, Sunchon National University) ;
• Varmazyar, Rezvan (Department of Mathematics, Khoy Branch, Islamic Azad University)
• Accepted : 2017.05.10
• Published : 2017.06.25

#### Abstract

All rings considered here are commutative rings with identity and all modules considered here are unital left modules. A submodule N of an R-module M is said to be extended to M if $N=aM$ for some ideal a of R and it is said to be fully invariant if ${\varphi}(L){\subseteq}L$ for every ${\varphi}{\in}End(M)$. An R-module M is called a [resp., fully invariant] multiplication module if every [resp., fully invariant] submodule is extended to M. The class of fully invariant multiplication modules is bigger than the class of multiplication modules. We deal with prime submodules and associated prime submodules of fully invariant multiplication modules. In particular, when M is a nonzero faithful multiplication module over a Noetherian ring, we characterize the zero-divisors of M in terms of the associated prime submodules, and we show that the set Aps(M) of associated prime submodules of M determines the set $Zdv_M(M)$ of zero-dvisors of M and the support Supp(M) of M.

#### Acknowledgement

Supported by : Sunchon National University

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