Quasi-hyperideals in multiplicative hyperrings

Abstract

A subring Q of a ring A is called a quasi-ideal of A if AQ intersection QA Q where AQ [QA] denotes the set of all finite sums of the form sigma aiqu[sigma qiai] where ai A and qi Q. Quasi-ideals are a generalization of left ideals and right ideals. Quasi-ideals in rings have long been studied and a lot of interesting theorems relating to quasi-ideals in rings have been provided. A hyperoperation on a nonempty set H is a function ๐ :H x H -> P*(H) where P(H) is the power set of H and P*(H) = P(H)\{is an empty set}. In this case, (H,๐) is called a hypergroupoid and for nonempty subsets X and Y of H, let X๐Y denote the union of all set x๐y where x and y run over X and Y, respectively. A semihypergroup is a hypergroupoid (H,๐) such that (x๐y)๐z = x๐(y๐z) for all x, y, z H. A multiplicative hyperring is a system (A, +, ๐) such that (i) (A, +) is an abelian group, (ii) (A, ๐) is a semihypergroup, (iii) x๐(y+z) x๐y + x๐z and (y+z)๐x y๐x + z๐x for all x, y, z A, (iv) (-x)๐y = x๐(-y) = -(x๐y) for all x, y A. If both containments in (iii) are equalities we say that (A, +, ๐) is strongly distributies. Subhyperrings, left [right] hyperideals, hyperideals and quasi-hyperideals of multiplicative hyperrings are similar in definitions to subrings, left [right] ideals, ideals and quasi-ideals of rings, respectively. We also have that quasi-hyperideals generalize left hyper-ideals and right hyperideals. Especially, quasi-hyperideals in multiplicative hyperrings and quasi-hyperideals in strongly distributive multiplicative hyperrings generalize quasi-ideals in rings. In this research, many well-known theorems on quasi-ideals in rings are generalized to theorems on quasi-hyperideals in multiplicative hyperrings or strongly distributive multiplicative hyperrings. Then those well-known facts in rings become our special cases.