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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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