Distinguished elements in semiring extensions

Abstract: In this paper, we investigate zero-divisor, nilpotent, idempotent, unit, small, and irreducible elements in semiring extensions such as amount, content, and monoid semialgebras. We also introduce new concepts such as the prime avoidance property in semirings, entire-like semirings, semialgebras with Property (A), and also, Armendariz and McCoy semialgebras and we prove some results related to these concepts. For example, we prove that if $B$ is an $S$-semialgebra, then under some conditions, the set of zero-divisors $Z(B)$ of $B$ is the union of the extended maximal primes of $Z(S)$. Finally, we prove a generalization of Eisenstein's irreducibility criterion.