Abstract
In this paper, graded rings are $S$-graded rings inducing $S,$ that is, rings whose additive groups can be written as a direct sum of a family of their additive subgroups indexed by a nonempty set $S,$ and such that the product of two homogeneous elements is again a homogeneous element. As a generalization of the recently introduced notion of a $UJ$-ring, we define a graded $UJ$-ring. Graded nil clean rings which are graded $UJ$ are described. We also investigate the graded $UJ$-property under some graded ring constructions.
Highlights
Independently, in [4,19], a U J-ring is introduced as an associative ring R with identity 1 such that 1+J(R) = U (R), where J(R) and U (R) denote the Jacobson radical of R and the group of units of R, respectively
We observe an S-graded ring inducing S [15,16,17], that is, a ring R whose additive group can be written as a direct sum of a family of additive subgroups of R indexed by a partial groupoid S, and such that RsRt ⊆ Rst whenever st is defined, and RsRt = 0 implies that st is defined
Let us recall, if R = s∈S Rs is an S-graded ring inducing S, it is said to be graded nil clean [13] if every homogeneous element of R can be written as a sum of a homogeneous idempotent and a homogeneous nilpotent element
Summary
Independently, in [4,19], a U J-ring is introduced as an associative ring R with identity 1 such that 1+J(R) = U (R), where J(R) and U (R) denote the Jacobson radical of R and the group of units of R, respectively As it is noticed in [19], a ring R is a U J-ring if and only if the set Q(R) of all quasi-regular elements of R coincides with J(R). Graded rings there are studied from the so-called homogeneous point of view by observing the homogeneous part of a graded ring with induced partial addition and everywhere defined multiplication. This approach goes back to [20].
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