A correction of the decomposability result in a paper by Meyer-Neutsch

TL;DR

This paper identifies a gap in a 1993 algebraic proof, confirms the theorem's validity, and provides a corrected proof using an extremal property of idempotents with broader implications.

Contribution

It corrects a proof in a previous work by Meyer and Neutsch and introduces a new extremal property of idempotents relevant to algebraic decomposability.

Findings

Theorem 11 is true despite the proof gap.

A corrected proof using extremal idempotent properties.

Enhanced understanding of algebraic decomposability.

Abstract

In this short note, it is shown that there is a gap in the proof of Theorem 11 in the paper of Meyer and Neutsch (J. of Algebra, 1993). We prove, nevertheless, that the statement of the theorem is true and fix the proof by using a certain extremal property of idempotents which has an independent interest.