Twisted planes

TL;DR

This paper introduces and characterizes a new family of twisted plane algebra structures on polynomial and power series modules over a commutative ring, expanding understanding of algebraic structures on these modules.

Contribution

It provides a novel characterization of twisted plane algebra structures on k[X,Y] and k[[X,Y]], generalizing previous results in the field.

Findings

New family of twisted plane algebra structures identified

Characterization of these structures on polynomial and power series modules

Extension of results to power series modules

Abstract

Let k be a commutative ring. We find and characterize a new family of twisted planes (i. e. associative unitary k-algebra structures on the k-module k[X,Y], having k[X] and and k[Y] as subalgebras).Similar results are obtained for the k-module of two variables power series k[[X,Y]].