Graded chain conditions and graded Jacobson radical of groupoid graded modules

Abstract: In this work, we continue to lay the groundwork for the theory of groupoid graded rings and modules. The main topics we address include graded chain conditions, the graded Jacobson radical, and the gr-socle for graded modules. We present several descending (ascending) chain conditions for graded modules and we refer to the most general one as $\Gamma_0$-artinian ($\Gamma_0$-noetherian). We show that $\Gamma_0$-artinian (resp. $\Gamma_0$-noetherian) modules share many properties with artinian (noetherian) modules in the classical theory. However, we present an example of a right $\Gamma_0$-artinian ring that is not right $\Gamma_0$-noetherian. Following the pattern of the classical case, we examine the basic properties of the graded Jacobson radical and the gr-socle for groupoid graded modules. We also establish some fundamental properties of the graded Jacobson radical of groupoid graded rings. Finally, we introduce the notion of gr-semilocal ring, which simultaneously generalizes the concepts of semilocal ring and semilocal (small) category.