An overring-theoretic approach to polynomial extensions of star and semistar operations

TL;DR

This paper investigates how semistar operations on polynomial domains extend from base integral domains, linking overring-theoretic properties with earlier stable semistar operation research to define canonical extensions.

Contribution

It introduces a general framework for extending semistar operations from an integral domain to its polynomial domain, connecting with prior stable semistar work.

Findings

Characterizes properties of semistar operation extensions on polynomial domains.

Links overring-theoretic approaches with stable semistar operations.

Provides conditions for canonical and strict extensions of semistar operations.

Abstract

Call a semistar operation $\ast$ on the polynomial domain $D[X]$ an extension (respectively, a strict extension) of a semistar operation $\star$ defined on an integral domain $D$, with quotient field $K$, if $E^\star = (E[X])^{\ast}\cap K$ (respectively, $E^\star [X]= (E[X])^{\ast}$) for all nonzero $D$-submodules $E$ of $K$. In this paper, we study the general properties of the above defined extensions and link our work with earlier efforts, centered on the stable semistar operation case, at defining semistar operations on $D[X]$ that are "canonical" extensions (or, "canonical" strict extensions) of semistar operations on $D$.