Some remarks on the comparability of ideals in semirings

A semiring is uniserial if its ideals are totally ordered by inclusion. First, we show that a semiring S is uniserial if and only if the matrix semiring M-n(S) 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 S are linearly ordered if and only if for each x, y is an element of S, there is a positive integer n such that either x vertical bar y(n )or y vertical bar x(n). Then, we introduce and characterize pseudo-valuat ion semidomains. It is shown that prime ideals of pseudovaluation semidomains and also of the divided ones are linearly ordered.