Semi-clean group rings

Abstract

Recall that a ring is said to be a clean ring if every element can be expressed as the sum of a unit and an idempotent . In one variant of this definition, a ring is said to be a semi-clean ring if every element can be expressed as the sum of a unit and a periodic element. Ye's Theorem [12] states that the group ring Z ( p ) [ C 3 ] is semi-clean, where p is a prime integer and C 3 is a cyclic group of order 3. In this article, we generalize Ye's Theorem by demonstrating that, if R is a local ring, then the group ring R [ G ] is semi-clean if and only if G is a torsion abelian group .