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Let G be a group and let S = g∈G Sg be a G-graded ring. Given a normal subgroup N of G, there is a naturally induced G/N -grading of S. It is well-known that if S is strongly G-graded, then the induced G/N -grading is strong for any N . The class of epsilon-strongly graded rings was recently introduced by Nystedt,Öinert and Pinedo as a generalization of unital strongly graded rings. We give an example of an epsilon-strongly graded partial skew group ring such that the induced quotient group grading is not epsilonstrong. Moreover, we give necessary and sufficient conditions for the induced G/N -grading of an epsilon-strongly G-graded ring to be epsilon-strong. Our method involves relating different types of rings equipped with local units (s-unital rings, rings with sets of local units, rings with enough idempotents) with generalized epsilon-strongly graded rings.

We provide a characterization of graded von Neumann regular rings involving the recently introduced class of nearly epsilon-strongly graded rings. As our main application, we generalize Hazrat's result that Leavitt path algebras over fields are graded von Neumann regular. More precisely, we show that a Leavitt path algebra LR(E) with coefficients in a unital ring R is graded von Neumann regular if and only if R is von Neumann regular. We also prove that both Leavitt path algebras and corner skew Laurent polynomial rings over von Neumann regular rings are semiprimitive and semiprime. Thereby, we generalize a result by Abrams and Aranda Pino on the semiprimitivity of Leavitt path algebras over fields.

Algebraic Cuntz-Pimsner rings are naturally Z-graded rings that generalize corner skew Laurent polynomial rings, Leavitt path algebras and unperforated Z-graded Steinberg algebras. In this article, we characterize strongly, epsilon-strongly and nearly epsilonstrongly Z-graded algebraic Cuntz-Pimsner rings up to graded isomorphism. We recover two results by Hazrat on when corner skew Laurent polynomial rings and Leavitt path algebras are strongly graded. As a further application, we characterize noetherian and artinian corner skew Laurent polynomial rings.

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