Wedderburn-Malcev decomposition of one-sided ideals of finite dimensional algebras

TL;DR

This paper proves a Wedderburn-Malcev type decomposition for one-sided ideals in finite dimensional associative algebras over perfect fields, splitting them into semisimple and radical parts.

Contribution

It establishes a decomposition of one-sided ideals into semisimple and radical components, extending classical results to a broader class of ideals.

Findings

Every one-sided ideal can be decomposed into semisimple and radical parts.

The decomposition involves a semisimple subalgebra and the radical of the algebra.

The result generalizes the Wedderburn-Malcev theorem to one-sided ideals.

Abstract

Let $A$ be a finite dimensional associative algebra over a perfect field and let $R$ be the radical of $A$. We show that for every one-sided ideal $I$ of $A$ there exists a semisimple subalgebra $S$ of $A$ such that $I=I_{S}\oplus I_{R}$ where $I_{S}=I\cap S$. and $I_{R}=I\cap R$.