A look at the prime and semiprime operations of one-dimensional domains

TL;DR

This paper investigates prime and semiprime operations in one-dimensional domains, showing the absence of bounded semiprime operations and characterizing prime operations in certain semigroup rings, with examples of non-identity prime operations.

Contribution

It demonstrates that no bounded semiprime operations exist on fractional ideals and identifies conditions under which prime operations are the identity, highlighting differences in rings with more generators.

Findings

No bounded semiprime operations on fractional ideals.

Prime operation is identity in semigroup rings with ≤2 generators.

Existence of non-identity prime operations in rings with ≥3 generators.

Abstract

We continue the analysis of prime and semiprime operations over one-dimensional domains started in \cite{Va}. We first show that there are no bounded semiprime operations on the set of fractional ideals of a one-dimensional domain. We then prove the only prime operation is the identity on the set of ideals in semigroup rings where the ideals are minimally generated by two or fewer elements. This is not likely the case in semigroup rings with ideals of three or more generators since we are able to exhibit that there is a non-identity prime operations on the set of ideals of $k[[t^3,t^4,t^5]]$.