Unique Factorisation in Semirings

Question

$\mathbb{N}$ forms a semiring with unique factorization (by the fundamental theorem of arithmetic) and only one unit (namely $1$). In other words, it is what could be considered as the equivalent of a UFD for semirings (is there an established name for such an object?). I presume that not every semiring is a semiring UFD, so here is my question: apart from $\mathbb{N}[x]$, what other semirings have unique factorization and exactly one unit?