In a ring, there is unique factorization domain. Then is there a similar concept in semiring - that is a commutative semiring that has unique factorization for every element except zero? If so, what would be such semiring that has infinitely many prime elements?
2026-04-29 18:15:58.1777486558
Semiring that has unique factorization except zero
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Well every UFD is already a semiring, but you are probably interested in an example that isn't a ring, so the obvious candidate is $\Bbb N=\{0,1,2,3,\ldots\}$.
If you are asking about what the definition of a unique factorization semiring would be, then I would say that the ring definition carries over pretty much word for word to semirings with no nonzero zero-divisors. The definition only makes reference to multiplication, after all.