Some results on retracts of polynomial rings

TL;DR

This paper investigates the structure of retracts of polynomial rings, focusing on invariants under group actions, general properties of retracts, and how ideals and ring properties behave under retractions.

Contribution

It provides new insights into the relationship between polynomial rings and their retracts, addressing open questions and exploring invariants and ideal behavior.

Findings

Characterization of invariants as retracts under ${ m G}_a$-actions

Analysis of properties of retracts in polynomial rings

Behavior of ideals under ring retractions

Abstract

In this paper, we first consider the relationship between a polynomial ring $B$ over a Noetherian domain $R$ and the ring of invariants $A$ of a ${\mathbb G}_a$-action on $B$, when $A$ occurs as a retract of $B$. Next, we study retracts of a polynomial ring in general and address the questions of D. L. Costa raised in \cite{C}. Finally, we examine the behaviour of ideals and certain properties of rings under retractions.