A CONSTRUCTION OF CONGRUENCE-SIMPLE SEMIRINGS

A construction of congruence-simple semirings is presented. The congruence-simple semirings of positive rational (real) numbers are fairly familiar, but the other (congruence-)simple semirings are regarded as somewhat apocryphal. It is easy to show that simple semirings split into three basic classes: the additively cancellative semirings, the additively nil-semirings and the additively idempotent ones. The first class includes all simple rings and many subsemirings of ordered rings. The second class includes many congruence-simple semigroups equipped with constant addition, but what remains is quite enigmatic so far ([1]). Now, we come to the third class, the additively idempotent simple semirings. These semirings (at least in the finite case) are of interest because of possible applications in cryptology (see e.g. [9]) and they are constructed as endomorphism semirings of semilattices (see [2], [6], [7] and [8]). The present note continues this line of research. Finally, notice that few pieces of information on simple semirings and general semirings are available in [4] or [5].