• Reduced Multiplication Modules

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      https://www.ias.ac.in/article/fulltext/pmsc/121/02/0121-0132

    • Keywords

       

      Multiplication module; reduced module; minimal prime submodule; Zariski topology; extremally disconnected.

    • Abstract

       

      An 𝑅-module 𝑀 is called a multiplication module if for each submodule 𝑁 of $M, N=IM$ for some ideal 𝐼 of 𝑅. As defined for a commutative ring 𝑅, an 𝑅-module 𝑀 is said to be reduced if the intersection of prime submodules of 𝑀 is zero. The prime spectrum and minimal prime submodules of the reduced module 𝑀 are studied. Essential submodules of 𝑀 are characterized via a topological property. It is shown that the Goldie dimension of 𝑀 is equal to the Souslin number of Spec $(M)$. Also a finitely generated module 𝑀 is a Baer module if and only if Spec $(M)$ is an extremally disconnected space; if and only if it is a $CS$-module. It is proved that a prime submodule 𝑁 is minimal in 𝑀 if and only if for each $x\in N,\mathrm{Ann}(x)\nsubseteq(N:M)$. When 𝑀 is finitely generated; it is shown that every prime submodule of 𝑀 is maximal if and only if 𝑀 is a von Neumann regular module ($VNM$); i.e., every principal submodule of 𝑀 is a summand submodule. Also if 𝑀 is an injective 𝑅-module, then 𝑀 is a $VNM$

    • Author Affiliations

       

      Karim Samei1

      1. Department of Mathematics, Islamic Azad University, Hamedan Branch, Iran
    • Dates

       
  • Proceedings – Mathematical Sciences | News

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