Let G be a monoid with identity e, and let R be a G-graded commutative ring. Graded weakly prime ideals in a G-graded commutative ring have been introduced and studied in [3]. Here we study graded weakly prime submodules of a G-graded R-module. A number of results concerning of these class of submodules are given. For example, we give some characterizations of homogeneous components of graded submodules.

A generalization of the original Diffie-Hellman key exchange in (Z/pZ) * found a new depth when Miller [10] and Koblitz [7] suggested that such a protocol could be used with the group over an elliptic curve. Maze, Monico and Rosenthal extend such a generalization to the setting of a semigroup action on a finite set, more precisely, linear actions of abelian semirings on semimodules [8]. In this paper, we extend such a generalization to the linear actions of quotient semirings on semimodules. In fact, we show how the action of a quotient semirings on a semimodule gives rise to a generalized Diffie-Hellman and ElGamal protocol. This leads naturally to a cryptographic protocol whose difficulty is based on the hardness of a particular control problem, namely the problem of steering the state of some dynamical system from an intial vector to some final location.

Abstract. Let I be a proper ideal of a commutative ring R with 1 = 0. The ideal-based zero-divisor graph of R with respect to I, denoted by ΓI (R), is the (simple) graph with vertices { x ∈ R \ I | xy ∈ I for some y ∈ R \ I }, and distinct vertices x and y are adjacent if and only if xy ∈ I. In this paper, we study ΓI (R) for commutative rings R such that R/I is a chained ring.

Let R be a commutative ring with identity and let M be a prime Rmodule. Let R(+)M be the idealization of the ring R by the R-module M . We study the diameter and girth of the zero-divisor graph of the ring R(+)M . Mathematics Subject Classification: 13A99, 13A15