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Simple commutative semirings. (English) Zbl 0976.16034
A commutative semiring \(S\) is congruence-simple if and only if one of the following six cases holds: (1) the two-element semirings (which are not listed below), (2) the semiring \(V(G)\) where \(G\) is a multiplicative Abelian group, \(V(G)=G\cup\{0\}\), \(0x=0=x0\), \(x+x=x\), \(x+y=0\) for \(x\neq y\), (3) the semiring \(W(A)\) for any subsemigroup \(A\) of the additive group \(\mathbb{R}(+)\) of the reals \(\mathbb{R}\) with addition \(a+b=\min(a,b)\) and multiplication \(a*b=a+b\) such that \(A\cap\mathbb{R}^+\neq\emptyset\neq A\cap\mathbb{R}^-\), (4) fields, (5) zero-multiplication rings of finite prime order, (6) certain subsemirings of the positive real numbers \(\mathbb{R}^+\) subject to certain constraints. Also ideal-simple commutative semirings, semifields, parasemifields and finitely generated simple semirings are discussed.

MSC:
16Y60 Semirings
12K10 Semifields
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