Suppose is a commutative unital ring and is an abelian group. We give a general criterion only in terms of and when all normalized units in the commutative group ring are -nilpotent. This extends recent results published in [Extracta Math., 2008–2009] and [Ann. Sci. Math. Québec, 2009].
An R-algebra A is called an E(R)-algebra if the canonical homomorphism from A to the endomorphism algebra of the R-module , taking any a ∈ A to the right multiplication by a, is an isomorphism of algebras. In this case is called an E(R)-module. There is a proper class of examples constructed in [4]. E(R)-algebras arise naturally in various topics of algebra. So it is not surprising that they were investigated thoroughly in the last decades; see [3, 5, 7, 8, 10, 13, 14, 15, 18, 19]. Despite...