A note on finitely generated ideal-simple commutative semirings. (English) Zbl 1192.16045

Kala, Vítězslav; Kepka, Tomáš

Commentat. Math. Univ. Carol. 49, No. 1, 1-9 (2008).

It is known that every infinite finitely generated congruence-simple (i.e., having exactly two congruences) commutative semiring is additively idempotent. On the other hand, the corresponding result seems to be open for ideal-simple (i.e., every ideal having more than one element is the whole semiring) commutative semirings.
In the paper, the authors reduce this problem to a special class of semirings. Namely, it is shown that every infinite finitely generated ideal-simple commutative semiring is additively idempotent iff every commutative parasemifield (i.e., semiring where the multiplicative semigroup is a non-trivial group) which is finitely generated as a semiring is additively idempotent.

Reviewer: Petr Němec (Praha)

MSC:

16Y60	Semirings
16D25	Ideals in associative algebras
12K10	Semifields

Keywords:

ideal-simple semirings; congruence-simple semirings; commutative semirings