Simple semirings with left multiplicatively absorbing elements (Q896239)

A \textit{semiring} is an algebraic structure \(S\) with two associative operations, usually denoted as addition (\(+\)) and multiplication (\(\cdot\)), where the addition is commutative and the multiplication distributes over the addition from both sides. An element \(w\in S\) is \textit{left} (resp.\ \textit{right}) \textit{multiplicatively absorbing} if \(wa=w\) (resp.\ \(aw=w\)) for all \(a\in S\). A semiring is \textit{non-trivial} if \(|S|\geq2\). A non-trivial semiring is called \textit{congruence-simple} if it has just two congruences. A commutative semigroup \((M,+)\) together with a scalar multiplication \(S\times M\to M\) is called a \textit{left \(S\)-semimodule} if \((a+b)x=ax+bx\), \(a(x+y)=ax+ay\) and \(a(bx)=(ab)x\) for all \(a,b\in S\) and \(x,y\in M\). An \(S\)-semimodule \(M\) is called \textit{faithful} if for all \(a,b\in S\), \(a\neq b\), there is at least one \(x\in M\) with \(ax\neq bx\). A non-trivial semiring \(S\) is of \textit{type} (A) if there is a faithful \(S\)-semimodule \(M\) such that for every \(x\in M\) there is an \(a\in S\) with \(aM=\{x\}\). Note that semirings of type (A) generalize the endomorphism semirings of non-trivial semilattices. In the paper under review, congruence-simple semirings with at least two left multiplicatively absorbing elements are investigated. By Theorem~3.5, all of them are of type (A). In their introduction, the authors state that ``congruence-simple semirings cause much more trouble than ideal-simple ones''. The main results of the paper give criteria (or necessary conditions) for a semiring of type (A) to be congruence-simple. The authors consider eight subtypes (A1)--(A8) of type (A). A typical result states that a semiring \(S\) of type (A) is congruence-simple if and only if either \(|S|=2\) or \(S\) is of types (A3) and (A4). The paper generalizes results from \textit{A. Kendziorra} and \textit{J. Zumbrägel} [J. Algebra 388, 43--64 (2013; Zbl 1286.16039)], where only finite semirings were considered.