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 [Formula: see text] is an [Formula: see text]-semialgebra, then under some conditions, the set of zero-divisors [Formula: see text] of [Formula: see text] is the union of the extended maximal primes of [Formula: see text]. Finally, we prove a generalization of Eisenstein’s irreducibility criterion.