Some properties of hypermodules over Krasner hyperrings

Abstract

A system (R,+,.) is said to be a Krasner hyperring if (i) (R,+) is a canonical hypergroup, (ii) (R,.) is a semigroup with zero 0 where 0 is the scalar identity of (R,+) and (iii) x.(y+z)=x.y+x.z and (y+z).x=y.x+z.x for all x,y,z [is an element of a set] R. A hypermodule over a Krasner hyperring R is a canonical hypergroup M, for which there is a function (r,m) [right arrow]rm from RxM into M such that for all r, r[subscript 1], r[subscript 2] [is an element of a set]R and m, m[subscript 1],m[subacript 2] [is an element of a set] M, (i) r(m[subscript 1]+m[subscript 2])=rm[subscript 1]+rm[subscript 2], (ii) )(r[subscript 1]+r[subscript 2])m=r[subscript 1]m+r[subscript 2]m , (iii) (r[subscript 1].r[subscript 2])m=r[subscript 1](r[subscript 2]m) and (iv) 0[subscript r]m=0[subscript M]. In this research, various elementary properties of modules over rings are generalized to properties of hypermodules over Krasner hyperrings and some concrete examples of hypermodules over Krasner hyperrings are given by considering among the collection of all multiplicative interval semigroups joining 0 on the system of real numbers and some hyperoperations. Moreover, we give a definition of projective hypermodule which is parallel to the definition of projective module in module theory and study some related properties.