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

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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.

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 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...

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

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...

We recall the notion of Hopf quasigroups introduced previously by the authors. We construct a bicrossproduct Hopf quasigroup $kM\u25b9\u25c2k\left(G\right)$ 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 ${G}_{\mathbb{O}}$ as transversal in an order 128 group $X$ with subgroup ${\mathbb{Z}}_{2}^{3}$ and hence obtain a Hopf quasigroup $k{G}_{\mathbb{O}}>\u25c2k\left({\mathbb{Z}}_{2}^{3}\right)$ as a particular case of our construction.

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...

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.