A Characterization of Commutative Group Rings Whose Units Are Feckly Trivial

Characterizing specific types of units in group rings in terms of the Jacobson radical constitutes a frequently investigated problem in the theory of group rings. In this study, since R is a commutative ring with unity and G is a finite Abelian group, Uft(RG) as the set of feckly trivial units in the group ring RG, consisting of units that are congruent modulo J(RG) to some g∈G, where J(RG) denotes the Jacobson radical of RG, is defined. Secondly, some necessary and sufficient conditions are given for the group U(RG) of units of the group ring RG to be feckly trivial under the assumption that supp(G)∩jp(R)=∅ where P is the set of all prime integers, supp(G)={p∈P:Gp≠eG}, jp(R)={p∈P:∃r∈R∖{0R},pr∈J(R)} and Gp is the p-primary component in G. Finally, two open problems related to this notion are introduced.