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arXiv · 1810.08836

Some remarks on the comparability of ideals in semirings

H. Behzadipour·P. Nasehpour
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Abstract

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 $x,y \in S$, there is a positive integer $n$ such that either $x|y^n$ or $y|x^n$. 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.

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H. Behzadipour, P. Nasehpour. 2022-03-11. Some remarks on the comparability of ideals in semirings. https://doi.org/10.1142/s1793557123500067

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