Hopkins-Levitzki Type Theorems for Groupoid Graded Rings

Abstract

We continue the study of the basic theory of object-unital groupoid graded rings. In this work, we are especially interested in nilpotency conditions on the graded Jacobson radical. We introduce the concept of left/right objectwise nilpotency of graded ideals, and prove that this condition is appropriate for obtaining graded generalizations of the Hopkins--Levitzki theorem. Although this condition is not symmetric, we show that its two-sided version is suitable for defining gr-semiprimary rings. It is known that one-sided $Γ_0$-artinian rings need not be $Γ_0$-noetherian, but using our tools we prove that one-sided gr-hereditary $Γ_0$-artinian rings, two-sided $Γ_0$-artinian rings, and $d$-finitely generated one-sided $Γ_0$-artinian rings are $Γ_0$-noetherian. However, the first class need not be gr-semiprimary, whereas the other two always are. We illustrate our results with several (counter)examples, especially involving graded upper triangular matrices.