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- Shahabaddin Ebrahimi Atani, Fatemeh Esmaeili, Khalil Saraei
- 2013

We introduce and investigate the total graph of a commutative semiring with non-zero identity. The main purpose of this paper is to extend the definition and some results given in [2] to a more general semiring case.

There is a natural graph associated to the zero-divisors of a commutative semiring with non-zero identity. In this article we essentially study zero-divisor graphs with respect to primal and non-primal ideals of a commutative semiring R and investigate the interplay between the semiring-theoretic properties of R and the graph-theoretic properties of ΓI(R)… (More)

Since the theory of ideals plays an important role in the theory of quotient semirings, in this paper, we will make an intensive study of the notions of Noetherian, Artinian, prime, primary, weakly primary and k-maximal ideals in commutative quotient semirings. The bulk of this paper is devoted to stating and proving analogues to several well-known theorems… (More)

Since the theory of ideals plays an important role in the theory of semirings, in this paper we will make an intensive study of the notions of primal and weakly primal ideals in commutative semirings with an identity 1. It is shown that these notions inherit most of the essential properties of the primal and weakly primal ideals of a commutative ring with… (More)

- Shahabaddin Ebrahimi Atani, Shokoofe Habibi
- 2011

The total graph of a commutative ring have been introduced and studied by D. F. Anderson and A. Badawi in [1]. In a manner analogous to a commutative ring, the total torsion element graph of a module M over a commtative ring R can be defined as the undirected graph T (Γ(M)). The basic properties and possible structures of the graph T (Γ(M)) are studied. The… (More)

In a manner analogous to a commutative ring, the idealbased zero-divisor graph of a commutative semiring R can be defined as the undirected graph ΓI(R) for some ideal I of R. The properties and possible structures of the graph ΓI (R) are studied.

- Shahabaddin Ebrahimi Atani, Ahamd Yousefian Darani, YOUSEFIAN DARANI
- 2009

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 ΓI(R) for some ideal I of R. Also we show that the zero-divisor graph with respect to primal… (More)

Let R be a commutative ring with non-zero identity. Various properties of multiplication modules are considered. We generalize Ohm’s properties for submodules of a finitely generated faithful multiplication Rmodule (see [8], [12] and [3]).

This paper establishes a set of theorems that describe the diameter of a zero-divisor graph for a finite direct product R1 × R2 × · · · × Rn with respect to the diameters of the zero-divisor graphs of R1, R2, · · · , Rn−1 and Rn(n > 2).