Displaying similar documents to “Inertial subrings of a locally finite algebra”

Commutativity of rings with constraints involving a subset

Moharram A. Khan (2003)

Czechoslovak Mathematical Journal

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Suppose that R is an associative ring with identity 1 , J ( R ) the Jacobson radical of R , and N ( R ) the set of nilpotent elements of R . Let m ≥ 1 be a fixed positive integer and R an m -torsion-free ring with identity 1 . The main result of the present paper asserts that R is commutative if R satisfies both the conditions (i) [ x m , y m ] = 0 for all x , y ∈ R ∖ J ( R ) and (ii) [ ( x y ) m + y m x m , x ] = 0 = [ ( y x ) m + x m y m , x ] , for all x , y ∈ R ∖ J ( R ) . This result is also valid if (i) and (ii) are replaced by (i) ' [ x m , y m ] = 0 for all x , y ∈ R ∖ N ( R ) and (ii) ' [ ( x y ) m + y m x m , x ] = 0 = [ ( y x ) m + x m y m , x ] for all x , y ∈ R ∖ N ( R ) . Other similar...

An example of a simple derivation in two variables

Andrzej Nowicki (2008)

Colloquium Mathematicae

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Let k be a field of characteristic zero. We prove that the derivation D = ∂ / ∂ x + ( y s + p x ) ( ∂ / ∂ y ) , where s ≥ 2, 0 ≠ p ∈ k, of the polynomial ring k[x,y] is simple.

Classification of rings satisfying some constraints on subsets

Moharram A. Khan (2007)

Archivum Mathematicum

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Let R be an associative ring with identity 1 and J ( R ) the Jacobson radical of R . Suppose that m ≥ 1 is a fixed positive integer and R an m -torsion-free ring with 1 . In the present paper, it is shown that R is commutative if R satisfies both the conditions (i) [ x m , y m ] = 0 for all x , y ∈ R ∖ J ( R ) and (ii) [ x , [ x , y m ] ] = 0 , for all x , y ∈ R ∖ J ( R ) . This result is also valid if (ii) is replaced by (ii)’ [ ( y x ) m x m - x m ( x y ) m , x ] = 0 , for all x , y ∈ R ∖ N ( R ) . Our results generalize many well-known commutativity theorems (cf. [1], [2], [3], [4], [5], [6], [9], [10], [11] and [14]). ...

Nil-clean and unit-regular elements in certain subrings of 𝕄 2 ( ℤ )

Yansheng Wu, Gaohua Tang, Guixin Deng, Yiqiang Zhou (2019)

Czechoslovak Mathematical Journal

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An element in a ring is clean (or, unit-regular) if it is the sum (or, the product) of an idempotent and a unit, and is nil-clean if it is the sum of an idempotent and a nilpotent. Firstly, we show that Jacobson’s lemma does not hold for nil-clean elements in a ring, answering a question posed by Koşan, Wang and Zhou (2016). Secondly, we present new counter-examples to Diesl’s question whether a nil-clean element is clean in a ring. Lastly, we give new examples of unit-regular elements...

CF-modules over commutative rings

Ahmed Najim, Mohammed Elhassani Charkani (2018)

Commentationes Mathematicae Universitatis Carolinae

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Let R be a commutative ring with unit. We give some criterions for determining when a direct sum of two CF-modules over R is a CF-module. When R is local, we characterize the CF-modules over R whose tensor product is a CF-module.

On rings all of whose modules are retractable

Şule Ecevit, Muhammet Tamer Koşan (2009)

Archivum Mathematicum

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Let R be a ring. A right R -module M is said to be retractable if 𝕋 H o m R ( M , N ) ≠ 0 whenever N is a non-zero submodule of M . The goal of this article is to investigate a ring R for which every right R-module is retractable. Such a ring will be called right mod-retractable. We proved that ( 1 ) The ring ∏ i ∈ ℐ R i is right mod-retractable if and only if each R i is a right mod-retractable ring for each i ∈ ℐ , where ℐ is an arbitrary finite set. ( 2 ) If R [ x ] is a mod-retractable ring then R is a mod-retractable ring.

Rings consisting entirely of certain elements

Huanyin Chen, Marjan Sheibani, Nahid Ashrafi (2018)

Czechoslovak Mathematical Journal

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We completely determine when a ring consists entirely of weak idempotents, units and nilpotents. We prove that such ring is exactly isomorphic to one of the following: a Boolean ring; ℤ 3 ⊕ ℤ 3 ; ℤ 3 ⊕ B where B is a Boolean ring; local ring with nil Jacobson radical; M 2 ( ℤ 2 ) or M 2 ( ℤ 3 ) ; or the ring of a Morita context with zero pairings where the underlying rings are ℤ 2 or ℤ 3 .