The Wedderburn–Artin Theorem for Paragraded Rings

Abstract

In this paper, we prove the paragraded version of the Wedderburn–Artin theorem. Following the methods known from the abstract case, we first prove the density theorem and observe the matrix rings whose entries are from a paragraded ring. However, in order to arrive at the desired structure theorem, we introduce the notion of a Jacobson radical of a paragraded ring and prove some properties which are analogous to the abstract case. In the process, we study the faithful and irreducible paragraded modules over noncommutative paragraded rings and prove the paragraded version of the well-known Schur lemma.