Euclidean Semirings

Abstract

If a is an element of a semiring R then we denote by RD(a) the set of all right divisors of a in the monoid (R, •). That is to say, (math). Since b ∈ RD(b) for all b ∈ R, it is clearly true that b ∈ RD(a) if and only if (math). Note that if R is a simple semiring and if b ∈ RD(a) then there exists an element r of R such that a = rb and so, by Proposition 4.3, we have a + b = rb + b = b. Thus we see that if a is an element of a simple semiring R then RD(a) &#x2260 Ø implies that a ∈ Z(R).