Simple semirings with zero. (Q2797002)

Let \((S,+,\cdot)\) be an additively commutative and idempotent semiring with the natural partial order \(x\leq y\Longleftrightarrow x+y=y\) for all \(x,y\in S\). A left ideal \(K\) of \(S\) is called complete if \(a\in K\) and \(b\leq a\) imply \(b\in K\). A left semimodule \(_SM\), clearly also assumed to be commutative and idempotent, is called faithful if for \(a\neq b\) in \(S\) there is some \(x\in M\) such that \(ax\neq bx\). A faithful left semimodule \(_SM\) is called characteristic if \((M,+)\) has a neutral element \(0\in M\) satisfying \(S0=\{0\}\), and there is a mapping \(\varepsilon\colon M\times M\to S\) such that \(\varepsilon(u,v)x=0\) and \(\varepsilon(u,v)y=v\) for all \(u,v,x,y\in M\), \(x\leq u\), \(y\nleq u\).NEWLINENEWLINE Now, let \((S,+,\cdot)\), \(|S|\geq 3\), have an absorbing zero as well as a greatest element. Assume further that for every complete left ideal \(K\) of \(S\) such that \(K\) and \(S\setminus K\) are infinite and there is a greatest element in \(K\). Then the following conditions are equivalent. (i) \(S\) is simple and has at least one minimal left ideal. (ii) \(S\) is simple and there is a faithful minimal semimodule \(_SN\). (iii) There exists some characteristic semimodule \(_SM\).