Narrow Results

We study a weak divisibility property for noncommutative rings: A nontrivial ring is fadelian if for all nonzero $a$ and $x$ there exist $b,c$ such that $x=ab+ca$. We prove properties of fadelian rings and construct examples thereof which are not division rings, as well as non-Noetherian and non-Ore examples.

The study of prime ideals has been an area of active research. In recent past a considerable work has been done in this direction. Associated prime ideals and minimal prime ideals of certain types of Ore extensions have been characterized.

In this paper a relation between completely prime ideals of a ring R and those of R[x; σ, δ] has been given; σ is an automorphisms of R and δ is a σ-derivation of R. It has been proved that if P is a completely prime ideal of R such that σ(P) = P and δ(P) ⊆ P, then P[x; σ, δ] is a completely prime ideal of R[x; σ, δ]. It has also been proved that this type of relation does not hold for strongly prime ideals.

The study of the prime ideals in Ore extension rings R[x,σ,δ] has attracted a lot of attention in recent years and has proven to be a challenging undertaking ([5], [7], [12], et al.). The present article makes a contribution to this study for the associated prime ideals. More precisely, we aim to describe how the associated primes of an R-module M_R behave under passage to the polynomial module M[x] over an Ore extension R[x,σ,δ]. If we impose natural σ-compatibility and δ-compatibility assumptions on the module M_R (see Sec. 2 below), we can describe all associated primes of the R[x,σ,δ]-module M[x] in terms of the associated primes of M_R in a very straightforward way. This result generalizes the author's recent work [1] on skew polynomial rings.

We classify the twisted tensor products of a finite set algebra with a two elements set algebra using colored quivers obtained through considerations analogous to Ore extensions. This provides also a classification of entwining structures between a finite set algebra and the grouplike coalgebra on two elements. The resulting 2-nilpotent algebras have particular features with respect to Hochschild (co)homology and cyclic homology.

A ring R is called (right principally) quasi-Baer if the right annihilator of every (principal right) ideal of R is generated by an idempotent. We study on the relationship between the quasi-Baer and p.q.-Baer property of a ring R and these of the Ore extension R[x; α, δ] for any automorphism α and α-derivation δ of R.

Let $R$ be an associative unital ring with an endomorphism $\alpha$ and $\alpha$-derivation $\delta$. Some types of ring elements such as the units and the idempotents play distinguished roles in noncommutative ring theory, and will play a central role in this work. In fact, we are interested to study the unit elements, the idempotent elements, the von Neumann regular elements, the $\pi$-regular elements and also the von Neumann local elements of the Ore extension ring $R[x;\alpha ,\delta ]$, when the base ring $R$ is a right duo ring which is $(\alpha ,\delta )$-compatible. As an application, we completely characterize the clean elements of the Ore extension ring $R[x;\alpha ,\delta ]$, when the base ring $R$ is a right duo ring which is $(\alpha ,\delta )$-compatible.

In this paper, we give a characterization of left (right) quasi-duo differential polynomial rings. In particular, we show that a differential polynomial ring is left quasi-duo if and only if it is right quasi-duo. This yields a partial answer to a question posed by Lam and Dugas in 2005. We provide nontrivial examples of such rings and give a complete description of the maximal ideals of an arbitrary quasi-duo differential polynomial ring. Moreover, we show that there is no left (right) quasi-duo differential polynomial ring in several indeterminates.

For a commutative integral domain $D$ with field of fractions $K$, the ring of integer-valued polynomials on $D$ is $\mathrm{\text{Int}}(D)=\{f\in K[x]|f(a)\in D\mathrm{\text{for all }}a\in D\}$. In this paper, we extend this construction to skew polynomial rings. Given an automorphism $\sigma$ of $K$, the skew polynomial ring $K[x;\sigma ]$ consists of polynomials with coefficients from $K$, and with multiplication given by $xa=\sigma (a)x$ for all $a\in K$. We define $\mathrm{\text{Int}}(D;\sigma )=\{f\in K[x;\sigma ]|f(a)\in D\mathrm{\text{for all}}a\in D\}$, which is the set of integer-valued skew polynomials on $D$. When $\sigma$ is not the identity, $K[x;\sigma ]$ is noncommutative and evaluation behaves differently than it does for ordinary polynomials. Nevertheless, we are able to prove that $\mathrm{\text{Int}}(D;\sigma )$ has a ring structure in many cases. We show how to produce elements of $\mathrm{\text{Int}}(D;\sigma )$ and investigate its properties regarding localization and Noetherian conditions. Particular attention is paid to the case where $D$ is a discrete valuation ring with finite residue field.

Let $\delta :R\to R$ be a derivation of a ring $R$, and let $T=R[t;\delta ]$ be the differential Ore extension of $R$ defined by putting $ta=at+\delta (a)$ for all $a\in R$. We describe the derivations of $T$ extending a derivation of $R$ and we deduce an explicit description of every derivation of $T$ in the case where $R$ is the polynomial ring $𝕜[x]$ over a field $𝕜$. As an application, we compute the first Hochschild cohomology group of $𝕜[x][t;\delta ]$ when $\delta =x^{m}d/dx$ and $m=1,2$; for $m>2$ a conjectural result is proposed.

Let $R$ be a ring, $\sigma$ be an automorphism of $R$ and $\delta$ be a $\sigma$-derivation of $R$. We use $\mathrm{\Omega }\subseteq \text{End}(R,+)$ to denote the set of all words composed of $\sigma$, ${\sigma }^{-1}$ and $\delta$. A $(\sigma ,{\sigma }^{-1},\delta )$-ideal $P$ of $R$ is $\mathrm{\Omega }$-prime if whenever $a,b\in R$ are such that $aR\omega (b)\subseteq P$ for any $\omega \in \mathrm{\Omega }$, we have $a\in P$ or $b\in P$. In this paper, we first introduce the $\mathrm{\Omega }$-prime ideal and the $\mathrm{\Omega }$-prime radical of a ring $R$, to obtain connections between the prime radical of the Ore extension $R[x;\sigma ,\delta ]$ and the $\mathrm{\Omega }$-prime radical of the base ring $R$. Based on these results, we next give definitions of the $\mathrm{\Omega }$-LS-prime ideal, the $\mathrm{\Omega }$-strongly prime ideal and the $\mathrm{\Omega }$-uniformly strongly prime ideal of a ring $R$ to provide formulas for the LS-prime radical, the strongly prime radical and the uniformly strongly prime radical of the Ore extension.

Bi-Koszul algebras, including two classes of non-Koszul Artin-Schelter regular algebras of global dimension 4, were a class of graded algebras with non-pure resolutions, introduced in [8]. There is a natural question: can we construct bi-Koszul algebras from algebras with pure resolutions? In this paper, we study this question in terms of normal extensions and Ore extensions. More precisely, we attempt to obtain bi-Koszul algebras from algebras with pure resolutions by these two kinds of extensions. Furthermore, some homological properties of bi-Koszul algebras obtained in such ways are discussed.

Let $R$ be a ring equipped with an endomorphism $\alpha$ and an $\alpha$-derivation $\delta$. In this article, we study $(\alpha ,\delta )$-constant Armendariz rings and radicals of the Ore extension $R[x;\alpha ,\delta ]$, in terms of a $(\alpha ,\delta )$-constant Armendariz ring $R$ with an $\alpha$-condition, are determined. We prove that several properties transfer between $R$ and the Ore extension $R[x;\alpha ,\delta ]$, in case $R$ is $\alpha$-compatible $(\alpha ,\delta )$-constant Armendariz.