Clean matrices over commutative rings

Huanyin Chen

Displaying similar documents to “Clean matrices over commutative rings”

A commutativity theorem for associative rings

Mohammad Ashraf (1995)

Archivum Mathematicum

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Let $m > 1, s \geq 1$ be fixed positive integers, and let $R$ be a ring with unity $1$ in which for every $x$ in $R$ there exist integers $p = p (x) \geq 0, q = q (x) \geq 0, n = n (x) \geq 0, r = r (x) \geq 0$ such that either $x^{p} [x^{n}, y] x^{q} = x^{r} [x, y^{m}] y^{s}$ or $x^{p} [x^{n}, y] x^{q} = y^{s} [x, y^{m}] x^{r}$ for all $y \in R$ . In the present paper it is shown that $R$ is commutative if it satisfies the property $Q (m)$ (i.e. for all $x, y \in R, m [x, y] = 0$ implies $[x, y] = 0$ ).

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...

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 $a \in R$ there exist two orthogonal idempotents $e, f \in R$ and a nilpotent $w \in R$ 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.

Unit-regularity and representability for semiartinian $*$ -regular rings

Christian Herrmann (2020)

Archivum Mathematicum

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We show that any semiartinian $*$ -regular ring $R$ is unit-regular; if, in addition, $R$ is subdirectly irreducible then it admits a representation within some inner product space.

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 $R \subseteq T$ is a ring extension such that $m T \subseteq R$ 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.

On stable equivalences of module subcategories over a semiperfect noetherian ring

Noritsugu Kameyama, Yuko Kimura, Kenji Nishida (2014)

Colloquium Mathematicae

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Given a semiperfect two-sided noetherian ring Λ, we study two subcategories $_{k} (Λ) = M \in m o d Λ | E x t_{Λ}^{j} (T r M, Λ) = 0 (1 \leq j \leq k)$ and $_{k} (Λ) = N \in m o d Λ | E x t_{Λ}^{j} (N, Λ) = 0 (1 \leq j \leq k)$ of the category mod Λ of finitely generated right Λ-modules, where Tr M is Auslander’s transpose of M. In particular, we give another convenient description of the categories $_{k} (Λ)$ and $_{k} (Λ)$ , and we study category equivalences and stable equivalences between them. Several results proved in [J. Algebra 301 (2006), 748-780] are extended to the case when Λ is a two-sided noetherian semiperfect ring.

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...

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; $ℤ_{3} \oplus ℤ_{3}$ ; $ℤ_{3} \oplus B$ 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 $ℤ_{3}$ .