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### A note on centralizers in $q$-deformed Heisenberg algebras.

AMA. Algebra Montpellier Announcements [electronic only]

### A note on Krull dimension of skew polynomial rings.

Lobachevskii Journal of Mathematics

### A note on locally inverse semigroup algebras.

International Journal of Mathematics and Mathematical Sciences

### A note on p.q.-Baer modules.

The New York Journal of Mathematics [electronic only]

### A Note on Skew Polynomial Rings

Publications de l'Institut Mathématique

### A note on the simplicity of skew polynomial rings of derivation type

Acta Mathematica Universitatis Ostraviensis

### A representation theorem for Chain rings

Colloquium Mathematicae

A ring A is called a chain ring if it is a local, both sided artinian, principal ideal ring. Let R be a commutative chain ring. Let A be a faithful R-algebra which is a chain ring such that Ā = A/J(A) is a separable field extension of R̅ = R/J(R). It follows from a recent result by Alkhamees and Singh that A has a commutative R-subalgebra R₀ which is a chain ring such that A = R₀ + J(A) and R₀ ∩ J(A) = J(R₀) = J(R)R₀. The structure of A in terms of a skew polynomial ring over R₀ is determined.

### A subclass of strongly clean rings

Communications in Mathematics

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 $a\in R$ is called very ${J}^{#}$-clean provided that there exists an idempotent $e\in R$ such that $ae=ea$ and $a-e$ or $a+e$ is an element of ${J}^{#}$. A ring $R$ is said to be very ${J}^{#}$-clean in case every element in $R$ is very ${J}^{#}$-clean. We prove that every very ${J}^{#}$-clean ring is strongly $\pi$-rad clean and has stable range one. It is shown...

### A unified approach to the Armendariz property of polynomial rings and power series rings

Colloquium Mathematicae

A ring R is called Armendariz (resp., Armendariz of power series type) if, whenever $\left({\sum }_{i\ge 0}{a}_{i}{x}^{i}\right)\left({\sum }_{j\ge 0}{b}_{j}{x}^{j}\right)=0$ in R[x] (resp., in R[[x]]), then ${a}_{i}{b}_{j}=0$ for all i and j. This paper deals with a unified generalization of the two concepts (see Definition 2). Some known results on Armendariz rings are extended to this more general situation and new results are obtained as consequences. For instance, it is proved that a ring R is Armendariz of power series type iff the same is true of R[[x]]. For an injective endomorphism σ of a ring...

### Abelian modules.

Acta Mathematica Universitatis Comenianae. New Series

### Algèbres de Bass et extensions de Ore itérées.

Collectanea Mathematica

We show that some iterated Ore extensions have the same behaviour with respect to injective resolutions as Gorenstein commutative rings.

Mathematica Pannonica

### An intermediate ring between a polynomial ring and a power series ring

Colloquium Mathematicae

Let R[x] and R[[x]] respectively denote the ring of polynomials and the ring of power series in one indeterminate x over a ring R. For an ideal I of R, denote by [R;I][x] the following subring of R[[x]]: [R;I][x]: = ${\sum }_{i\ge 0}{r}_{i}{x}^{i}\in R\left[\left[x\right]\right]$ : ∃ 0 ≤ n∈ ℤ such that ${r}_{i}\in I$, ∀ i ≥ n. The polynomial and power series rings over R are extreme cases where I = 0 or R, but there are ideals I such that neither R[x] nor R[[x]] is isomorphic to [R;I][x]. The results characterizing polynomial rings or power series rings with a certain ring...

### Associated prime ideals of skew polynomial rings.

Beiträge zur Algebra und Geometrie

### Automorphisms and f-simplicity in skew polynomial rings

Δελτίο της Ελληνικής Μαθηματικής Εταιρίας

### Bicrossproduct Hopf quasigroups

Commentationes Mathematicae Universitatis Carolinae

We recall the notion of Hopf quasigroups introduced previously by the authors. We construct a bicrossproduct Hopf quasigroup $kM▹◂k\left(G\right)$ from every group $X$ with a finite subgroup $G\subset X$ and IP quasigroup transversal $M\subset X$ subject to certain conditions. We identify the octonions quasigroup ${G}_{𝕆}$ as transversal in an order 128 group $X$ with subgroup ${ℤ}_{2}^{3}$ and hence obtain a Hopf quasigroup $k{G}_{𝕆}>◂k\left({ℤ}_{2}^{3}\right)$ as a particular case of our construction.

### Bimodule resolution of the Liu-Schulz algebras.

Zapiski Nauchnykh Seminarov POMI

### Binomial Skew Polynomial Rings, Artin-Schelter Regularity, and Binomial Solutions of the Yang-Baxter Equation

Serdica Mathematical Journal

2000 Mathematics Subject Classification: Primary 81R50, 16W50, 16S36, 16S37.Let k be a field and X be a set of n elements. We introduce and study a class of quadratic k-algebras called quantum binomial algebras. Our main result shows that such an algebra A defines a solution of the classical Yang-Baxter equation (YBE), if and only if its Koszul dual A! is Frobenius of dimension n, with a regular socle and for each x, y ∈ X an equality of the type xyy = αzzt, where α ∈ k {0, and z, t ∈ X is satisfied...

### Categorifications of the polynomial ring

Fundamenta Mathematicae

We develop a diagrammatic categorification of the polynomial ring ℤ[x]. Our categorification satisfies a version of Bernstein-Gelfand-Gelfand reciprocity property with the indecomposable projective modules corresponding to xⁿ and standard modules to (x-1)ⁿ in the Grothendieck ring.

### Central Armendariz rings.

Bulletin of the Malaysian Mathematical Sciences Society. Second Series

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