Related papers: Derivations and Iterated Skew Polynomial Rings

We introduce the class E2 (resp. SE2) of commutative rings R with the property that each unimodular 2 x 2 matrix with entries in R extends to an invertible 3 x 3 matrix (resp. invertible 3 x 3 matrix whose (3, 3) entry is 0). Among…

Over a finite field $\mathbb{F}_{q^m}$, the evaluation of skew polynomials is intimately related to the evaluation of linearized polynomials. This connection allows one to relate the concept of polynomial independence defined for skew…

Let R be a Stanley-Reisner ring (that is, a reduced monomial ring) with coefficients in a domain k, and K its associated simplicial complex. Also let D_k(R) be the ring of k-linear differential operators on R. We give two different…

Let $R$ be a commutative ring with identity and $S$ a multiplicative subset of $R$. First, we introduce and study the $S$-projective dimensions and $S$-injective dimensions of $R$-modules, and then explore the $S$-global dimension…

In this paper, we introduce and study two new classes of commutative rings, namely semi transitional rings and transitional rings, which extend several classical ideas arising from rings of continuous functions and their variants. A general…

The matrix exponential restricted to skew-symmetric matrices has numerous applications, notably in view of its interpretation as the Lie group exponential and Riemannian exponential for the special orthogonal group. We characterize the…

Let $R$ be a prime ring of characteristic different from $2$, $U$ be its Utumi quotient ring, $C$ be its extended centroid and $L$ be a non-central Lie ideal of $R$. Suppose $F$ and $G$ are two non-zero $b$-generalized skew derivations of…

Let $\Bbbk$ be an algebraically closed field of characteristic $0$. In this paper, we study the Grothendieck ring $G_0(D(H_\mathcal{D}))$ and the projective class ring $r_p(D(H_\mathcal{D}))$ of the Drinfeld double $D(H_{\mathcal{D}})$ of…

The first purpose of this paper is to set up a general notion of skew power series rings S over a coefficient ring R, which are then studied by filtered ring techniques. The second subject consists of investigating the class of S-modules…

We show that the formal skew Laurent series ring $R = D(\! ( x; \sigma )\! )$ over a commutative Dedekind domain $D$ with an automorphism $\sigma$ is a noncommutative Dedekind domain. If $\sigma$ acts trivially on the ideal class group of…

In this paper, we consider five possible extensions of the Pr\"ufer domain notion to the case of commutative rings with zero-divisors. We investigate the transfer of these Pr\"ufer-like properties between a ring $R$ and $R\bowtie I$; his…

Recollements of triangulated categories may be seen as exact sequences of such categories. Iterated recollements of triangulated categories are analogues of geometric or topological stratifications and of composition series of algebraic…

The aim of the paper is to extend the class of generalized Weyl algebras to a larger class of rings (they are also called {\em generalized Weyl algebras}) that are determined by two ring endomorphisms rather than one as in the case of `old'…

Let $R$ be a commutative ring and $g(t) \in R[t]$ a monic polynomial. The commutative ring of polynomials $f(C_g)$ in the companion matrix $C_g$ of $g(t)$, where $f(t)\in R[t]$, is called the Companion Ring of $g(t)$. Special instances…

Let I be a finitely supported complete m-primary ideal of a regular local ring (R, m). A theorem of Lipman implies that I has a unique factorization as a *-product of special *-simple complete ideals with possibly negative exponents for…

This paper introduces and studies a new class of rings called {\it $U\sqrt{\Delta}$-rings}. A ring $R$ is $U\sqrt{\Delta}$ if every non-unit element can be written as the product of a unit and an element from $\sqrt{\Delta(R)}$, where…

This study explores in-depth the structure and properties of the so-called {\it strongly $\Delta$-clean rings}, that is a novel class of rings in which each ring element decomposes into a sum of a commuting idempotent and an element from…

We determine criteria for the prime spectrum of an ambiskew polynomial algebra $R$ over an algebraically closed field $K$ to be akin to those of two of the principal examples of such an algebra, namely the universal enveloping algebra…

The paper studies the question of existence of polynomials with given roots over associative non-commutative rings with identity. It is shown that in the case of an associative division ring for arbitrary n elements of this ring there…

Let $K$ be a field of arbitrary characteristic, $\CA$ be a commutative $K$-algebra which is a domain of essentially finite type (eg, the algebra of functions on an irreducible affine algebraic variety), $\ga_r$ be its {\em Jacobian ideal},…