Every simple compact semiring is finite. (Q281741)

The authors study compact Hausdorff topological semirings (not necessarily commutative). Such a semiring \(S\) is \textit{simple} if every non-zero continuous homomorphism \(f\colon S\to R\) to another Hausdorff topological semiring \(R\) is injective, which is shown to be equivalent to the condition that \(S\) has no non-trivial closed congruences.NEWLINENEWLINE The main theorem states that each simple compact semiring is finite, generalizing a classical result of \textit{I. Kaplansky} [Am. J. Math. 69, 153-183 (1947; Zbl 0034.16604)] that concerns finiteness of simple compact rings. Among other things, the proof uses properties of the (pre-)order \(\leq\) on \(S\) defined by \(a\leq b\) if \(a+c=b\) for some \(c\), and of congruences generated by subtractive ideals.