Semidistributive Laurent Series Rings

TL;DR

This paper characterizes when skew Laurent series rings are right semidistributive, showing that such properties in the Laurent series ring imply similar properties in the base ring, especially under semilocal and Artinian conditions.

Contribution

It establishes necessary and sufficient conditions for Laurent series rings to be right semidistributive, linking their properties to those of the base ring.

Findings

Skew Laurent series rings are right semidistributive iff the base ring is right semidistributive and Artinian.

The Laurent series ring $A((x))$ is right semidistributive and semilocal iff $A$ is right semidistributive and Artinian.

The study extends understanding of semidistributivity in noncommutative Laurent series rings.

Abstract

If $A$ is a ring with automorphism $\varphi$ and the skew Laurent series ring $A((x,\varphi ))$ is a right semidistributive semilocal ring then $A$ is a right semidistributive right Artinian ring. The Laurent series ring $A((x))$ is a right semidistributive semilocal ring if and only if $A$ is a right semidistributive right Artinian ring. The study is supported by Russian Scientific Foundation (project 16-11-10013).