• Karim Samei

      Articles written in Proceedings – Mathematical Sciences

    • Reduced Multiplication Modules

      Karim Samei

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      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 $VNM$

    • Comparison of graphs associated to a commutative Artinian ring


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      Let $R$ be a commutative ringwith $1 \neq 0$ and the additive group $R^{+}$. Several graphs on $R$ have been introduced by many authors, among zero-divisor graph $\Gamma_{1}(R)$, co-maximal graph $\Gamma_{2}(R)$, annihilator graph $AG(R)$, total graph $T (\Gamma(R))$, cozero-divisors graph $\Gamma_{c}(R)$, equivalence classes graph $\Gamma_{E}(R)$ and the Cayley graph Cay$(R^{+}, Z^{\ast}(R))$.Shekarriz et al. (J. Commun. Algebra, 40 (2012) 2798–2807) gave some conditions under which total graph is isomorphic to Cay$(R^{+}, Z^{\ast}(R))$. Badawi (J. Commun. Algebra, 42 (2014) 108–121) showed that when $R$ is a reduced ring, the annihilator graph is identical to the zero-divisor graph if and only if $R$ has exactly two minimal prime ideals. The purpose of this paper is comparison of graphs associated to a commutative Artinian ring. Among the results,we prove that for a commutative finite ring $R$ with $|Max(R)| = n \geq 3$, $\Gamma_{1}(R) \simeq \Gamma_{2}(R)$ if and only if $R \simeq \mathbb{Z}^{n}_{2}$; if and only if $\Gamma_{1}(R) \simeq \Gamma_{E}(R)$. Also then annihilator graph is identical to the cozero-divisor graph if and only if $R$ is a Frobenius ring.

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