Some Properties of Multiplicative -Rings of Polynomials over Multiplicative Hyperrings

The set of all polynomials , over a multiplicative hyperring , form a commutative group with respect to the component-wise addition (+) of the polynomials. For polynomials in , is a set of polynomials whose th components are chosen from the set , where and are the th and the th components of and , respectively. A multiplicative hyperring is polynomially structured if the hyperstructure is a multiplicative -ring. The purpose of the paper is to study the properties of the multiplicative -ring , corresponding to those of a polynomially structured multiplicative hyperring . 1. Introduction -structures [1] are introduced by Vougiouklis, at the Fourth AHA congress in the year of 1990. Since then, the study of -structure theory has been approached in several directions by many researchers (see [2–5]). The essence of the notion of -structures is to generalize the well-known algebraic hyperstructures (such as hypergroup, hyperring, and hypermodule), simply by replacing some or all axioms of the respective hyperstructures by the corresponding weak axioms. The -structure of our initial concern is multiplicative -ring, studied in [6, 7], which is a commutative group along with a hyperoperation such that (i) is an -semigroup [3, 8] (i.e., a hyperstructure in which is weak associative in the sense that , for all ) and (ii) is weak distributive with respect to + (i.e., and , for all . A multiplicative -ring is commutative if , for all . The identity element of the group is said to be absorbing in the multiplicative -ring if , for all . A nonempty finite subset of a multiplicative -ring is called an identity set (or -set, in short) [9] of if (i) for at least one and (ii) for any , . An element of is called a hyperidentity of if the set is an -set of . Unlike a ring, the equality of the set-expressions , , and does not hold in general on a multiplicative -ring for any . In fact, if is the ring of integers and if is a hyperoperation on , defined by , for all , then is a commutative multiplicative -ring, in which , , and . A multiplicative -ring is said to satisfy the condition () if the set equality (called the condition () [9]) holds true for any two elements and of . Let be a ring and a hyperoperation on , defined by , for all . Then, is a multiplicative -ring with condition (). We consider now an -structure in which (i) is a commutative group, (ii) is an -semigroup, and (iii) is semidistributive across the operation + (i.e., and , for all ). This -structure is clearly a multiplicative -ring and we thus call it a semidistributive multiplicative -ring. Henceforth,