We extend Dade's theorem to the realm of nearly epsilon-strongly graded rings, and present
new characterizations of strongly and epsilon-strongly graded rings. Further, we give …

In this article we generalize a classical result by Passman on primeness of unital strongly
group graded rings to the class of nearly epsilon-strongly group graded rings which are not …

There are two established gradings on Leavitt path algebras associated with ultragraphs,
namely the grading by the integers group and the grading by the free group on the edges. In …

Let R be a unital ring, let E be a directed graph and recall that the Leavitt path algebra LR
(E) carries a natural ℤ-gradation. We show that LR (E) is strongly ℤ-graded if and only if E is …

Let G be a group and let S=⊕ g∈ GS g 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 …

The three famous problems concerning units, zero-divisors and idempotents in group rings
of torsion-free groups, commonly attributed to Kaplansky, have been around for more than …

Let $ S $ be a partial groupoid, that is, a set with a partial binary operation. An $ S $-graded
ring $ R $ is said to be graded von Neumann regular if $ x\in xRx $ for every homogeneous …

Let G be a group with neutral element e and let S=⊕ g∈ GS g S=⨁_g∈GS_g be a G-
graded ring. A necessary condition for S to be noetherian is that the principal component S e …

Let G be a group, let A=⊕ g∈ GA g be an epsilon-strongly graded ring over G, let R:= A 1 be
the homogeneous component associated with the identity of G, and let 𝙿𝚒𝚌𝚂⁢(R) be the …

The results that are stated in P. Nystedt and J. Öinert [Group gradations on Leavitt path
algebras, J. Algebra Appl. 19 (9)(2020) 2050165, Sec. 4] hold true, but due to an …