Polynomial extensions of semistar operations

TL;DR

This paper provides a comprehensive method for extending semistar operations from an integral domain to its polynomial ring, and explores the behavior of classical operations under these extensions.

Contribution

It offers a complete solution for extending any semistar operation to polynomial rings and analyzes classical operations' behavior in this context.

Findings

Complete extension method for semistar operations to polynomial rings

Reinterpretation of previous results for specific semistar operations

Analysis of classical operations like $d_D$, $v_D$, $t_D$, $w_D$, and $b_D$ in polynomial extensions

Abstract

We provide a complete solution to the problem of extending arbitrary semistar operations of an integral domain $D$ to semistar operations of the polynomial ring $D[X]$. As an application, we show that one can reobtain the main results of some previous papers concerning the problem in the special cases of stable semistar operations of finite type or semistar operations defined by families of overrings. Finally, we investigate the behavior of the polynomial extensions of the most important and classical operations such as $d_D$, $v_D$, $t_D$, $w_D$ and $b_D$ operations.