A Note on Radicals and Polynomial Rings.
Let be an associative ring with identity and let denote the Jacobson radical of . is said to be semilocal if is Artinian. In this paper we give necessary and sufficient conditions for the group ring , where is an abelian group, to be semilocal.
The paper was motivated by Kovacs’ paper (1973), Isaacs’ paper (1980) and a recent paper, due to Brešar et al. (2018), concerning Skolem-Noether algebras. Let be a unital commutative ring, not necessarily a field. Given a unital -algebra , where is contained in the center of , , the goal of this paper is to study the question: when can a homomorphism be extended to an inner automorphism of ? As an application of main results presented in the paper, it is proved that if is a semilocal...