Some remarks on the comparability of ideals in semirings

Abstract

A semiring is uniserial if its ideals are totally ordered by inclusion. First, we show that a semiring [Formula: see text] is uniserial if and only if the matrix semiring [Formula: see text] is uniserial. As a generalization of valuation semirings, we also investigate those semirings whose prime ideals are linearly ordered by inclusion. For example, we prove that the prime ideals of a commutative semiring [Formula: see text] are linearly ordered if and only if for each [Formula: see text], there is a positive integer [Formula: see text] such that either [Formula: see text] or [Formula: see text]. Then, we introduce and characterize pseudo-valuation semidomains. It is shown that prime ideals of pseudo-valuation semidomains and also of the divided ones are linearly ordered.