Semigroups admitting skew-ring or skew-semifield structures

Abstract

A semiring is a system (S, +, .) such that (S, +) and (S, .) are semigroups and . is distributive over +. By a skew-ring we mean a semiring (S, +, .) such that (S, +) is a group. An element 0 of a semiring S = (S, +, .) is a zero of S if x + 0 = 0 + x = x and 0 . x = x . 0 = 0 for all x S. A skew-semifield is an additively commutative semiring (S, +, .) with zero 0 such that (S\{0}, .) is a group. For a semigroup S, the semigroup S0 is defined to be S if S has a zero and S contains more than one element, otherwise, let S0 be the semigroup S with a zero 0 adjoined. A semigroup S is said to admit a skew-ring structure if there exists an operation + on S0 such that (S0, +, .) is a skew-ring where . is the operation on S0. A group admitting skew-semifield structure is defined similary. Let R be a commutative ring with identity 1 = 0, Mn(R) the semigroup of all n ด n matrices over R under matrix multiplication and Gn(R) the subgroup of Mn(R) consisting ofall invertible n ด n matrices over R. Let V be a vector space over a division ring, L(V) the semigroup under composition of all linear transformations a : V --> V and G(V) the subgroup of L(V) consisting of all isomorphisms a : V --> V. In this research, various subsemigroups of Mn(R) and L(V) are characterized when they admit a skew-ring structure. Many subgroups of Gn(R) and G(V) are considered. We give characterizations determining when they admit a skew-semifield structure.