A semiring S is said to be a quasi completely regular semiring if for any a ∈ S there exists a positive integer n such that na is completely regular. The present paper is devoted to the study of completely Archimedean semirings. We show that a semiring S is a completely Archimedean semiring if and only if it is a nil-extension of a completely simple semiring. This result extends the crucial structure theorem of completely Archimedean semigroup.

Parasemifields (i.e., commutative semirings whose multiplicative semigroups are groups) are considered in more detail. We show that if a parasemifield $S$ contains ${\mathbb{Q}}^{+}$ as a subparasemifield and is generated by ${\mathbb{Q}}^{+}\cup \left\{a\right\}$, $a\in S$, as a semiring, then $S$ is (as a semiring) not finitely generated.