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Atani, S. Ebrahimi,Khoramdel, M.,Hesari, S. Dolati Pish Department of Mathematics 2014 Kyungpook mathematical journal Vol.54 No.2
The purpose of this paper is to introduce the concept of strongly extending modules which are particular subclass of the class of extending modules, and study some basic properties of this new class of modules. A module M is called strongly extending if each submodule of M is essential in a fully invariant direct summand of M. In this paper we examine the behavior of the class of strongly extending modules with respect to the preservation of this property in direct summands and direct sums and give some properties of these modules, for instance, strongly summand intersection property and weakly co-Hopfian property. Also such modules are characterized over commutative Dedekind domains.
TOTAL GRAPH OF A COMMUTATIVE SEMIRING WITH RESPECT TO IDENTITY-SUMMAND ELEMENTS
Atani, Shahabaddin Ebrahimi,Hesari, Saboura Dolati Pish,Khoramdel, Mehdi Korean Mathematical Society 2014 대한수학회지 Vol.51 No.3
Let R be an I-semiring and S(R) be the set of all identity-summand elements of R. In this paper we introduce the total graph of R with respect to identity-summand elements, denoted by T(${\Gamma}(R)$), and investigate basic properties of S(R) which help us to gain interesting results about T(${\Gamma}(R)$) and its subgraphs.
THE IDENTITY-SUMMAND GRAPH OF COMMUTATIVE SEMIRINGS
Atani, Shahabaddin Ebrahimi,Hesari, Saboura Dolati Pish,Khoramdel, Mehdi Korean Mathematical Society 2014 대한수학회지 Vol.51 No.1
An element r of a commutative semiring R with identity is said to be identity-summand if there exists $1{\neq}a{\in}R$ such that r+a = 1. In this paper, we introduce and investigate the identity-summand graph of R, denoted by ${\Gamma}(R)$. It is the (undirected) graph whose vertices are the non-identity identity-summands of R with two distinct vertices joint by an edge when the sum of the vertices is 1. The basic properties and possible structures of the graph ${\Gamma}(R)$ are studied.
MULTIPLICATION MODULES OVER PULLBACK RINGS (I)
ATANI, SHAHABADDIN EBRAHIMI,LEE, SANG CHEOL 호남수학회 2006 호남수학학술지 Vol.28 No.1
First, we give a complete description of the multiplication modules over local Dedekind domains. Second, if R is the pullback ring of two local Dedekind domains over a common factor field then we give a complete description of separated multiplication modules over R.
TOTAL IDENTITY-SUMMAND GRAPH OF A COMMUTATIVE SEMIRING WITH RESPECT TO A CO-IDEAL
Atani, Shahabaddin Ebrahimi,Hesari, Saboura Dolati Pish,Khoramdel, Mehdi Korean Mathematical Society 2015 대한수학회지 Vol.52 No.1
Let R be a semiring, I a strong co-ideal of R and S(I) the set of all elements of R which are not prime to I. In this paper we investigate some interesting properties of S(I) and introduce the total identity-summand graph of a semiring R with respect to a co-ideal I. It is the graph with all elements of R as vertices and for distinct x, $y{\in}R$, the vertices x and y are adjacent if and only if $xy{\in}S(I)$.
STRONGLY IRREDUCIBLE SUBMODULES
ATANI, SHAHABADDIN EBRAHIMI Korean Mathematical Society 2005 대한수학회보 Vol.42 No.1
This paper is motivated by the results in [6]. We study some properties of strongly irreducible submodules of a module. In fact, our objective is to investigate strongly irreducible modules and to examine in particular when sub modules of a module are strongly irreducible. For example, we show that prime submodules of a multiplication module are strongly irreducible, and a characterization is given of a multiplication module over a Noetherian ring which contain a non-prime strongly irreducible submodule.
ZERO-DIVISOR GRAPHS WITH RESPECT TO PRIMAL AND WEAKLY PRIMAL IDEALS
Atani, Shahabaddin Ebrahimi,Darani, Ahamd Yousefian Korean Mathematical Society 2009 대한수학회지 Vol.46 No.2
We consider zero-divisor graphs with respect to primal, nonprimal, weakly prime and weakly primal ideals of a commutative ring R with non-zero identity. We investigate the interplay between the ringtheoretic properties of R and the graph-theoretic properties of ${\Gamma}_I(R)$ for some ideal I of R. Also we show that the zero-divisor graph with respect to primal ideals commutes by localization.