Trivial units in commutative group algebras

Abstract

Let 𝐺 be an arbitrary abelian group and let 𝑅 be a commutative unitary ring of arbitrary characteristic. A necessary and su±cient condition is given for when all units in the group ring 𝑅𝐺 are trivial provided that either supp(𝐺) ∩ inv(𝑅) ≠∅ ; or 𝑅𝐺 is modular. In particular, we establish a comprehensive characterization by finding a criterion when 𝑅𝐺 has only trivial units provided that char(𝑅) is a positive number greater than 1. These achievements strengthen results due to Karpilovsky (Arch. Math. Basel, 1983), Herman-Li Parmenter (Can. Math. Bull., 2005) and the author (Math. Commun., 2005).