On graded nil clean rings

Ilić-Georgijević, Emil; Şahinkaya, Serap

doi:10.1080/00927872.2018.1435791

Cited by 8 publications

(12 citation statements)

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“…Next we want to characterize graded U J-rings which are graded nil clean. Let us recall, if R = s∈S R s is an S-graded ring inducing S, then 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. Therefore, if R is graded nil clean, it follows that every R e , where e ∈ E(S), is a nil clean ring.…”

Section: Graded U J-ringsmentioning

confidence: 99%

“…We give basic properties of graded U J-rings and discuss their connection to rings introduced in [13], namely to graded nil clean rings. More precisely, we characterize graded nil clean graded U J-rings.…”

Section: Introductionmentioning

confidence: 99%

“…By Theorem 2.2, the e-component of R/J g (R) is R e /J(R e ). According to the proof of Theorem 3.27 in[13] (see also Theorem 3.2 in[11]), we have that every nonzero homogeneous element from R/J g (R) can be identified with exactly one…”

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confidence: 97%

“…If R is graded nil clean, then J g (R) is a graded-nil ideal of R. Let us just mention that Lemma 3.28 in[13] assumes the existence of identity in a ring in order to use the fact that idempotents lift modulo nil ideals of a ring with identity. However, according to[18], the same holds true for rings without an identity.…”

mentioning

confidence: 99%

“…However, according to[18], the same holds true for rings without an identity. Moreover, let us mention that Lemma 3.28 in[13] says that R is graded nil clean if and only if R/I is graded nil clean provided that I is a graded-nil ideal of R and that the grading set is cancellative. Then the following statements are equivalent: i) R is a graded clean general graded U J-ring with the graded-nil graded Jacobson radical J g (R);…”

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confidence: 99%

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On Graded Uj-Rings

Ilić-Georgijević¹

2020

International Electronic Journal of Algebra

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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 U J-ring, we define a graded U J-ring. Graded nil clean rings which are graded U J are described. We also investigate the graded U J-property under some graded ring constructions.

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Section: Graded U J-ringsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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confidence: 97%

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On Graded Uj-Rings

Ilić-Georgijević¹

2020

International Electronic Journal of Algebra

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Graded Cancellation Properties of Graded Rings and Graded Unit-regular Leavitt Path Algebras

Vaš

2020

Algebr Represent Theor

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We raise the following general question regarding a ring graded by a group: "If P is a ring-theoretic property, how does one define the graded version P gr of the property P in a meaningful way?". Some properties of rings have straightforward and unambiguous generalizations to their graded versions and these generalizations satisfy all the matching properties of the nongraded case. If P is either being unit-regular, having stable range 1 or being directly finite, that is not the case. The first half of the paper addresses this issue. Searching for appropriate generalizations, we consider graded versions of cancellation, internal cancellation, substitution, and module-theoretic direct finiteness.In the second half of the paper, we turn to Leavitt path algebras. For Leavitt path algebras of finite graphs, we characterize graded unit-regularity and other cancellation properties in terms of the graph properties. Then we provide a complete description of graded matrix algebras over a trivially graded field which are graded isomorphic to Leavitt path algebras. As a consequence, we show that there are graded corners of Leavitt path algebras which are not graded isomorphic to Leavitt path algebras. This contrasts a recent result stating that every corner of a Leavitt path algebra of a finite graph is isomorphic to another Leavitt path algebra. If R is a finite direct sum of graded matricial algebras over a trivially graded field and over naturally graded fields of Laurent polynomials, we also present conditions under which R can be realized as a Leavitt path algebra.

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