# direct product decompositions of semi-rings

(With Hilbert's Basis Theorem in mind.)

Is it true that every finitely presented commutative semi-ring with unit is a finite direct product of directly indecomposable factors?

• How do you define a finitely presented semiring? – rschwieb May 28 at 17:40
• @rschwieb: A semiring is f.p. if it is the quotient of a finitely generated free semiring by a finitely generated congruence. Maybe `finitely presentable' is better? Anyway, it should be correct analogue of the ring case. – Boogie May 28 at 17:57
• How nice is the theory of idempotents for semirings? It looks like the main character $1-e$ gets dealt a blow since we don't have additive inverses... – rschwieb May 28 at 18:04
• @rschwieb A pair of elements $x, y$ such that ${x + y = 1}$ and ${x y = 0}$ certainly determines a direct product decomposition. But I think more is required to answer the question. And I would have guessed that the answer is known, but I have not found it in the web. – Boogie May 28 at 18:13