Euclidean Semirings

Jonathan S. Golan
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Jonathan S. Golan: University of Haifa

Chapter 12 in Semirings and their Applications, 1999, pp 135-141 from Springer

Abstract: 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) ≠ Ø implies that a ∈ Z(R).

Date: 1999
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DOI: 10.1007/978-94-015-9333-5_12

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