Displaying similar documents to “On weakened ( α , δ ) -skew Armendariz rings”

Notes on generalizations of Bézout rings

Haitham El Alaoui, Hakima Mouanis (2021)

Commentationes Mathematicae Universitatis Carolinae

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In this paper, we give new characterizations of the P - 2 -Bézout property of trivial ring extensions. Also, we investigate the transfer of this property to homomorphic images and to finite direct products. Our results generate original examples which enrich the current literature with new examples of non- 2 -Bézout P - 2 -Bézout rings and examples of non- P -Bézout P - 2 -Bézout rings.

P-injective group rings

Liang Shen (2020)

Czechoslovak Mathematical Journal

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A ring R is called right P-injective if every homomorphism from a principal right ideal of R to R R can be extended to a homomorphism from R R to R R . Let R be a ring and G a group. Based on a result of Nicholson and Yousif, we prove that the group ring RG is right P-injective if and only if (a) R is right P-injective; (b) G is locally finite; and (c) for any finite subgroup H of G and any principal right ideal I of RH , if f ∈ Hom R ( I R , R R ) , then there exists g ∈ Hom R ( RH R , R R ) such that g | I = f . Similarly, we also obtain equivalent...

(Generalized) filter properties of the amalgamated algebra

Yusof Azimi (2022)

Archivum Mathematicum

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Let R and S be commutative rings with unity, f : R → S a ring homomorphism and J an ideal of S . Then the subring R ⋈ f J : = { ( a , f ( a ) + j ) ∣ a ∈ R and j ∈ J } of R × S is called the amalgamation of R with S along J with respect to f . In this paper, we determine when R ⋈ f J is a (generalized) filter ring.

Semicommutativity of the rings relative to prime radical

Handan Kose, Burcu Ungor (2015)

Commentationes Mathematicae Universitatis Carolinae

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In this paper, we introduce a new kind of rings that behave like semicommutative rings, but satisfy yet more known results. This kind of rings is called P -semicommutative. We prove that a ring R is P -semicommutative if and only if R [ x ] is P -semicommutative if and only if R [ x , x - 1 ] is P -semicommutative. Also, if R [ [ x ] ] is P -semicommutative, then R is P -semicommutative. The converse holds provided that P ( R ) is nilpotent and R is power serieswise Armendariz. For each positive integer n , R is P -semicommutative...

About G-rings

Najib Mahdou (2017)

Commentationes Mathematicae Universitatis Carolinae

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In this paper, we are concerned with G-rings. We generalize the Kaplansky’s theorem to rings with zero-divisors. Also, we assert that if R ⊆ T is a ring extension such that m T ⊆ R for some regular element m of T , then T is a G-ring if and only if so is R . Also, we examine the transfer of the G-ring property to trivial ring extensions. Finally, we conclude the paper with illustrative examples discussing the utility and limits of our results.

Strongly 2-nil-clean rings with involutions

Huanyin Chen, Marjan Sheibani Abdolyousefi (2019)

Czechoslovak Mathematical Journal

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A * -ring R is strongly 2-nil- * -clean if every element in R is the sum of two projections and a nilpotent that commute. Fundamental properties of such * -rings are obtained. We prove that a * -ring R is strongly 2-nil- * -clean if and only if for all a ∈ R , a 2 ∈ R is strongly nil- * -clean, if and only if for any a ∈ R there exists a * -tripotent e ∈ R such that a - e ∈ R is nilpotent and e a = a e , if and only if R is a strongly * -clean SN ring, if and only if R is abelian, J ( R ) is nil and R / J ( R ) is * -tripotent. Furthermore, we explore...

A subclass of strongly clean rings

Orhan Gurgun, Sait Halicioglu and Burcu Ungor (2015)

Communications in Mathematics

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In this paper, we introduce a subclass of strongly clean rings. Let R be a ring with identity, J be the Jacobson radical of R , and let J # denote the set of all elements of R which are nilpotent in R / J . An element a ∈ R is called provided that there exists an idempotent e ∈ R such that a e = e a and a - e or a + e is an element of J # . A ring R is said to be in case every element in R is very J # -clean. We prove that every very J # -clean ring is strongly π -rad clean and has stable range one. It is shown that for a...

Symmetric and reversible properties of bi-amalgamated rings

Antonysamy Aruldoss, Chelliah Selvaraj (2024)

Czechoslovak Mathematical Journal

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Let f : A → B and g : A → C be two ring homomorphisms and let K and K ' be two ideals of B and C , respectively, such that f - 1 ( K ) = g - 1 ( K ' ) . We investigate unipotent, symmetric and reversible properties of the bi-amalgamation ring A ⋈ f , g ( K , K ' ) of A with ( B , C ) along ( K , K ' ) with respect to ( f , g ) .

Avoidance principle and intersection property for a class of rings

Rahul Kumar, Atul Gaur (2020)

Czechoslovak Mathematical Journal

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Let R be a commutative ring with identity. If a ring R is contained in an arbitrary union of rings, then R is contained in one of them under various conditions. Similarly, if an arbitrary intersection of rings is contained in R , then R contains one of them under various conditions.

A generalization of reflexive rings

Mete Burak Çalcı, Huanyin Chen, Sait Halıcıoğlu (2024)

Mathematica Bohemica

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We introduce a class of rings which is a generalization of reflexive rings and J -reversible rings. Let R be a ring with identity and J ( R ) denote the Jacobson radical of R . A ring R is called J -reflexive if for any a , b ∈ R , a R b = 0 implies b R a ⊆ J ( R ) . We give some characterizations of a J -reflexive ring. We prove that some results of reflexive rings can be extended to J -reflexive rings for this general setting. We conclude some relations between J -reflexive rings and some related rings. We investigate some extensions...