Multiplicative cancellativity of semirings and semigroups

Abstract

In this note, the usual ring-like definition of a multiplicatively cancellative semiring S= (S, +, 9 ) (cf. [1], [2]) and some generalizations of this concept will be clarified by equivalent characterizations. As a matter of fact, the intrinsic parts of these considerations are consequences of statements concerning merely the multiplicative semigroup (S, .) of S, regarded as an arbitrary one, and our results on semirings become those on very general algebras (S, +, .). In this context, we give some applications on nearrings.A semiring S=(S, +, .) is defined as an algebra such that (S, +) and (S, .) are any semigroups, connected by ring-like distributivity. A neutral element o of (S, +), if there is one, is called the zero of the semiring S, and in particular an annihilating zero iff, as for rings, {o} is an ideal of (S, 9 ). For simplicity, expressions like "ar or "S\{o}" are used for any semiring S, being of no consequence if S has no zero.We further denote by 13 [9t] the set of all elements of S being left [right] cancellable in (S, .). Clearly, both are subsemigroups of (S, .) if not empty, and for all x, yES we have (1) xyE13 =" yE13; xyE91 =* xE91.In general, very few can be said about the multiplicative behaviour of a zero of a semiring S (of. [2]). So it is notable that, as claimed in [4], the ring-like concept of multiplicative cancellability (in short: a#o=~aE 13 fq 91), applied to any semiring S, forces a zero of S to be either multiplicatively cancellable, too, or else annihilating. But even corresponding one-sided or mixed versions of this (replacing 13(q91 by !3, 91, or 13U 91) will prove correct, and we start with left and two-sided cancellability, treated simultaneously. To avoid trivial rubs, always ISl=>2 is assumed. DEFINITION 1. A semiring S is called multiplieatively (left) caneellative iff each element ar of S is (left) cancellable in (S, .). PROPOSITION 1..4 semiring S is multiplieatively (left) eaneellative iff one of the following conditions holds: ~) S has no zero, and (S, .) is (left) eaneellative. [1) S has a zero, and (S, .) is (left) eancellative. 7) S has an annihilating zero, and (S',,,{o}, .) is a subsemigroup of (S, .), which in (left) eaneellative?i With obvious definitions, we have for each semiring S: (S~{o}, .) is a subsemigroup ~. S has no proper divisors of zero .~ {o} is a completely prime set. But even for semirings with an annihilating zero this does not imply anything concerning cancellability. We note in this context (cf. [2]): If a E S is (left) cancellable in (S, .), a is no (left) zero divisor, but no conversely.