Geodesic
A note on finitely generated ideal-simple commutative semirings
Commentationes Mathematicae Universitatis Carolinae, Tome 49 (2008) no. 1, pp. 1-9.

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Many infinite finitely generated ideal-simple commutative semirings are additively idempotent. It is not clear whether this is true in general. However, to solve the problem, one can restrict oneself only to parasemifields.
Classification : 16Y60
Keywords: semiring; ideal; simple 
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Kala, Vítězslav; Kepka, Tomáš. A note on finitely generated ideal-simple commutative semirings. Commentationes Mathematicae Universitatis Carolinae, Tome 49 (2008) no. 1, pp. 1-9. http://geodesic.mathdoc.fr/item/CMUC_2008__49_1_a0/