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

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.

MSC:

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