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- 저자 : Rezvan Varmazyar (검색결과 10 건)
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On graded $(m, n)$-closed submodules
Rezvan Varmazyar 대한수학회 2023 대한수학회논문집 Vol.38 No.4
Let $A$ be a $G$-graded commutative ring with identity and $M$ a graded $A$-module. Let $m, n$ be positive integers with $m>n$. A proper graded submodule $L$ of $M$ is said to be graded $(m, n)$-closed if $a^{m}_g\cdot x_t\in L$ implies that $a^{n}_g\cdot x_t\in L$, where $a_g\in h(A)$ and $x_t\in h(M)$. The aim of this paper is to explore some basic properties of these class of submodules which are a generalization of graded $(m, n)$-closed ideals. Also, we investigate $GC^{m}_n-rad$ property for graded submodules.
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ANNIHILATOR GRAPHS OF COMMUTATOR POSETS
( Rezvan Varmazyar ) 호남수학회 2018 호남수학학술지 Vol.40 No.1
Let P be a commutator poset with Z(P) its set of zero-divisors. The annihilator graph of P, denoted by AG(P), is the (undirected) graph with all elements of Z(P) | {0} as vertices, and distinct vertices x, y are adjacent if and only if ann(xy) ≠ ann(x) ∪ ann(y). In this paper, we study basic properties of AG(P).
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ANNIHILATOR GRAPHS OF COMMUTATOR POSETS
Varmazyar, Rezvan The Honam Mathematical Society 2018 호남수학학술지 Vol.40 No.1
Let P be a commutator poset with Z(P) its set of zero-divisors. The annihilator graph of P, denoted by AG(P), is the (undirected) graph with all elements of $Z(P){\setminus}\{0\}$ as vertices, and distinct vertices x, y are adjacent if and only if $ann(xy)\;{\neq}\;(x)\;{\cup}\;ann(y)$. In this paper, we study basic properties of AG(P).
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Semiprime submodules of graded multiplication modules
이상철,Rezvan Varmazyar 대한수학회 2012 대한수학회지 Vol.49 No.2
Let G be a group. Let R be a G-graded commutative ring with identity and M be a G-graded multiplication module over R. A proper graded submodule Q of M is semiprime if whenever InK ⊆ Q,where I ⊆ h(R), n is a positive integer, and K h(M), then IK⊆Q. We characterize semiprime submodules of M. For example, we show that a proper graded submodule Q of M is semiprime if and only if grad(Q) ∩ h(M) = Q ∩ h(M). Furthermore if M is finitely generated,then we prove that every proper graded submodule of M is contained in a graded semiprime submodule of M. A proper graded submodule Q of M is said to be almost semiprime if (grad(Q) ∩ h(M))n(grad(0M) ∩ h(M))= (Q ∩ h(M))n(grad(0M) ∩ Q ∩ h(M)):Let K, Q be graded submodules of M. If K and Q are almost semiprime in M such that Q + K ̸= M and Q ∩ K Mg for all g 2 G, then we prove that Q + K is almost semiprime in M.
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ZERO-DIVISOR GRAPHS OF MULTIPLICATION MODULES
이상철,Rezvan Varmazyar 호남수학회 2012 호남수학학술지 Vol.34 No.4
In this study, we investigate the concept of zero-divisor graphs of multiplication modules over commutative rings as a nat-ural generalization of zero-divisor graphs of commutative rings. In particular, we study the zero-divisor graphs of the module Zn over the ring Z of integers, where n is a positive integer greater than 1.
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CHROMATIC NUMBER OF THE ZERO-DIVISOR GRAPHS OVER MODULES
Lee, Sang Cheol,Varmazyar, Rezvan Korean Mathematical Society 2019 대한수학회논문집 Vol.34 No.2
Let R be a commutative ring with identity and let M be an R-module. The main purpose of this paper is to calculate the chromatic number of the zero-divisor graphs over modules.
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SEMIPRIME SUBMODULES OF GRADED MULTIPLICATION MODULES
Lee, Sang-Cheol,Varmazyar, Rezvan Korean Mathematical Society 2012 대한수학회지 Vol.49 No.2
Let G be a group. Let R be a G-graded commutative ring with identity and M be a G-graded multiplication module over R. A proper graded submodule Q of M is semiprime if whenever $I^nK{\subseteq}Q$, where $I{\subseteq}h(R)$, n is a positive integer, and $K{\subseteq}h(M)$, then $IK{\subseteq}Q$. We characterize semiprime submodules of M. For example, we show that a proper graded submodule Q of M is semiprime if and only if grad$(Q){\cap}h(M)=Q+{\cap}h(M)$. Furthermore if M is finitely generated then we prove that every proper graded submodule of M is contained in a graded semiprime submodule of M. A proper graded submodule Q of M is said to be almost semiprime if (grad(Q)$\cap$h(M))n(grad$(0_M){\cap}h(M)$) = (Q$\cap$h(M))n(grad$(0_M){\cap}Q{\cap}h(M)$). Let K, Q be graded submodules of M. If K and Q are almost semiprime in M such that Q + K $\neq$ M and $Q{\cap}K{\subseteq}M_g$ for all $g{\in}G$, then we prove that Q + K is almost semiprime in M.
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ZERO-DIVISOR GRAPHS OF MULTIPLICATION MODULES
Lee, Sang Cheol,Varmazyar, Rezvan The Honam Mathematical Society 2012 호남수학학술지 Vol.34 No.4
In this study, we investigate the concept of zero-divisor graphs of multiplication modules over commutative rings as a natural generalization of zero-divisor graphs of commutative rings. In particular, we study the zero-divisor graphs of the module $\mathbb{Z}_n$ over the ring $\mathbb{Z}$ of integers, where $n$ is a positive integer greater than 1.
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ASSOCIATED PRIME SUBMODULES OF A MULTIPLICATION MODULE
Lee, Sang Cheol,Song, Yeong Moo,Varmazyar, Rezvan The Honam Mathematical Society 2017 호남수학학술지 Vol.39 No.2
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.
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ASSOCIATED PRIME SUBMODULES OF A MULTIPLICATION MODULE
( Sang Cheol Lee ),( Yeong Moo Song ),( Rezvan Varmazyar ) 호남수학회 2017 호남수학학술지 Vol.39 No.2
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 φ(L) ⊆ L for every φ ∈ 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<sub>M</sub>(M) of zero-dvisors of M and the support Supp(M) of M.
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