Reduced Multiplication Modules

Proceedings Mathematical Sciences

Karim Samei¹ 

[1] Department of Mathematics, Islamic Azad University, Hamedan Branch, Iran$$

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

DOI  :  

学科分类：数学（综合）

来源: Indian Academy of Sciences

【 摘 要 】

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 $xin N,mathrm{Ann}(x)subseteq(N:M)$. 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 $VNM$

【 授权许可】

Unknown