Displaying similar documents to “Nil-clean and unit-regular elements in certain subrings of 𝕄 2 ( ℤ ) ”

On feebly nil-clean rings

Marjan Sheibani Abdolyousefi, Neda Pouyan (2024)

Czechoslovak Mathematical Journal

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A ring R is feebly nil-clean if for any a ∈ R there exist two orthogonal idempotents e , f ∈ R and a nilpotent w ∈ R such that a = e - f + w . Let R be a 2-primal feebly nil-clean ring. We prove that every matrix ring over R is feebly nil-clean. The result for rings of bounded index is also obtained. These provide many classes of rings over which every matrix is the sum of orthogonal idempotent and nilpotent matrices.

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 .

Fundamental relation onm-idempotent hyperrings

Morteza Norouzi, Irina Cristea (2017)

Open Mathematics

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The γ*-relation defined on a general hyperring R is the smallest strongly regular relation such that the quotient R/γ* is a ring. In this note we consider a particular class of hyperrings, where we define a new equivalence, called [...] εm∗ ε m * , smaller than γ* and we prove it is the smallest strongly regular relation on such hyperrings such that the quotient R/ [...] εm∗ ε m * is a ring. Moreover, we introduce the concept of m-idempotent hyperrings, show that they are a characterization for...

Rings in which elements are sum of a central element and an element in the Jacobson radical

Guanglin Ma, Yao Wang, André Leroy (2024)

Czechoslovak Mathematical Journal

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An element in a ring R is called CJ if it is of the form c + j , where c belongs to the center and j is an element from the Jacobson radical. A ring R is called CJ if each element of R is CJ. We establish the basic properties of CJ rings, give several characterizations of these rings, and connect this notion with many standard elementwise properties such as clean, uniquely clean, nil clean, CN, and CU. We study the behavior of this notion under various ring extensions. In particular, we show...

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.

Skew inverse power series rings over a ring with projective socle

Kamal Paykan (2017)

Czechoslovak Mathematical Journal

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A ring R is called a right PS -ring if its socle, Soc ( R R ) , is projective. Nicholson and Watters have shown that if R is a right PS -ring, then so are the polynomial ring R [ x ] and power series ring R [ [ x ] ] . In this paper, it is proved that, under suitable conditions, if R has a (flat) projective socle, then so does the skew inverse power series ring R [ [ x - 1 ; α , δ ] ] and the skew polynomial ring R [ x ; α , δ ] , where R is an associative ring equipped with an automorphism α and an α -derivation δ . Our results extend and unify many existing...

A minimal regular ring extension of C(X)

M. Henriksen, R. Raphael, R. G. Woods (2002)

Fundamenta Mathematicae

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Let G(X) denote the smallest (von Neumann) regular ring of real-valued functions with domain X that contains C(X), the ring of continuous real-valued functions on a Tikhonov topological space (X,τ). We investigate when G(X) coincides with the ring C ( X , τ δ ) of continuous real-valued functions on the space ( X , τ δ ) , where τ δ is the smallest Tikhonov topology on X for which τ ⊆ τ δ and C ( X , τ δ ) is von Neumann regular. The compact and metric spaces for which G ( X ) = C ( X , τ δ ) are characterized. Necessary, and different sufficient,...

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...

Left EM rings

Jongwook Baeck (2024)

Czechoslovak Mathematical Journal

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Let R [ x ] be the polynomial ring over a ring R with unity. A polynomial f ( x ) ∈ R [ x ] is referred to as a left annihilating content polynomial (left ACP) if there exist an element r ∈ R and a polynomial g ( x ) ∈ R [ x ] such that f ( x ) = r g ( x ) and g ( x ) is not a right zero-divisor polynomial in R [ x ] . A ring R is referred to as left EM if each polynomial f ( x ) ∈ R [ x ] is a left ACP. We observe the structure of left EM rings with various properties, and study the relationships between the one-sided EM condition and other standard ring theoretic conditions....

Note on strongly nil clean elements in rings

Aleksandra Kostić, Zoran Z. Petrović, Zoran S. Pucanović, Maja Roslavcev (2019)

Czechoslovak Mathematical Journal

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Let R be an associative unital ring and let a ∈ R be a strongly nil clean element. We introduce a new idea for examining the properties of these elements. This approach allows us to generalize some results on nil clean and strongly nil clean rings. Also, using this technique many previous proofs can be significantly shortened. Some shorter proofs concerning nil clean elements in rings in general, and in matrix rings in particular, are presented, together with some generalizations of these...

Automorphisms of completely primary finite rings of characteristic p

Chiteng'a John Chikunji (2008)

Colloquium Mathematicae

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A completely primary ring is a ring R with identity 1 ≠ 0 whose subset of zero-divisors forms the unique maximal ideal . We determine the structure of the group of automorphisms Aut(R) of a completely primary finite ring R of characteristic p, such that if is the Jacobson radical of R, then ³ = (0), ² ≠ (0), the annihilator of coincides with ² and R / ≅ G F ( p r ) , the finite field of p r elements, for any prime p and any positive integer r.

On matrix Lie rings over a commutative ring that contain the special linear Lie ring

Evgenii L. Bashkirov, Esra Pekönür (2016)

Commentationes Mathematicae Universitatis Carolinae

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Let K be an associative and commutative ring with 1 , k a subring of K such that 1 ∈ k , n ≥ 2 an integer. The paper describes subrings of the general linear Lie ring g l n ( K ) that contain the Lie ring of all traceless matrices over k .

On weakened ( α , δ ) -skew Armendariz rings

Alireza Majdabadi Farahani, Mohammad Maghasedi, Farideh Heydari, Hamidagha Tavallaee (2022)

Mathematica Bohemica

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In this note, for a ring endomorphism α and an α -derivation δ of a ring R , the notion of weakened ( α , δ ) -skew Armendariz rings is introduced as a generalization of α -rigid rings and weak Armendariz rings. It is proved that R is a weakened ( α , δ ) -skew Armendariz ring if and only if T n ( R ) is weakened ( α ¯ , δ ¯ ) -skew Armendariz if and only if R [ x ] / ( x n ) is weakened ( α ¯ , δ ¯ ) -skew Armendariz ring for any positive integer n .

On some noetherian rings of C ∞ germs on a real closed field

Abdelhafed Elkhadiri (2011)

Annales Polonici Mathematici

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Let R be a real closed field, and denote by R , n the ring of germs, at the origin of Rⁿ, of C ∞ functions in a neighborhood of 0 ∈ Rⁿ. For each n ∈ ℕ, we construct a quasianalytic subring R , n ⊂ R , n with some natural properties. We prove that, for each n ∈ ℕ, R , n is a noetherian ring and if R = ℝ (the field of real numbers), then ℝ , n = ₙ , where ₙ is the ring of germs, at the origin of ℝⁿ, of real analytic functions. Finally, we prove the Real Nullstellensatz and solve Hilbert’s 17th Problem for the ring R , n . ...