Related papers: On quasi-Baer rings of Ore extensions

In this note we answer two questions on quasi-Baer modules raised by Lee and Rizvi in J.Algebra (2016).

Motivated by the construction of new examples of Artin-Schelter regular algebras of global dimension four, J.J. Zhang and J. Zhang (2008) introduced an algebra extension $A_P[y_1,y_2;\sigma,\delta,\tau]$ of $A$, which they called a double…

Suppose $F$ is a field with valuation $v$ and valuation ring $O_{v}$, $E$ is a finite field extension and $w$ is a quasi-valuation on $E$ extending $v$. We study quasi-valuations on $E$ that extend $v$; in particular, their corresponding…

In this note, we investigate the Baer splitting problem over commutative rings. In particular, we show that if a commutative ring $R$ is $\tau_q$-semisimple, then every Baer $R$-module is projective.

We introduce non-associative Ore extensions, $S = R[X ; \sigma , \delta]$, for any non-associative unital ring $R$ and any additive maps $\sigma,\delta : R \rightarrow R$ satisfying $\sigma(1)=1$ and $\delta(1)=0$. In the special case when…

Let R = D[x;\sigma;\delta] be an Ore extension over a commutative Dedekind domain D, where \sigma is an automorphism on D. In the case \delta = 0 Marubayashi et. al. already investigated the class of minimal prime ideals in term of their…

Let $R$ be a commutative ring with identity. The ring $R\times R$ can be viewed as an extension of $R$ via the diagonal map $\Delta: R \hookrightarrow R\times R$, given by $\Delta(r) = (r, r)$ for all $r\in R$. It is shown that, for any $a,…

We show that there exist noncommutative Ore extensions in which every right ideal is two-sided. This answers a problem posed by Marks in Duo Rings and Ore extensions, J.Algebra 280(2), (2004). We also provide an easy construction of one…

Let $B$ be a Poisson algebra $\Bbb C[x_1,\ldots, x_k]$ with Poisson bracket such that $$\{x_j,x_i\}=c_{ji}x_ix_j+p_{ji}$$ for all $j>i$, where $c_{ji}\in\Bbb C$ and $p_{ji}\in\Bbb C[x_1,\ldots,x_i]$. Here we obtain an iterated skew…

Let $R$ be a ring with identity, $(S,\leq)$ an ordered monoid, $\omega:S \to End(R)$ a monoid homomorphism, and $A= R\left[\left[S,\omega \right]\right]$ the ring of skew generalized power series. The concepts of generalized Baer and…

The following extension of Bohr's theorem is established: If a somewhere convergent Dirichlet series $f$ has an analytic continuation to the half-plane $\mathbb{C}_\theta = \{s = \sigma+it\,:\, \sigma>\theta\}$ that maps $\mathbb{C}_\theta$…

A real matrix $Q$ is quasi-orthogonal if $Q^{\top}Q=qI$, for some positive real number $q$. We prove that any $n\times n$ skew-symmetric matrix $S$ is a principal sub-matrix of a skew-symmetric quasi-orthogonal matrix $Q$, called a…

A characterization of right (left) quasi-duo skew polynomial rings of endomorphism type and skew Laurent polynomial rings are given. In particular, it is shown that (1) the polynomial ring R[x] is right quasi-duo iff R[x] is commutative…

We define and explore the bounded skew power series ring $R^+[[x;\sigma,\delta]]$ defined over a complete, filtered, Noetherian prime ring $R$ with a commuting skew derivation $(\sigma,\delta)$. We establish precise criteria for when this…

A description of right (left) quasi-duo Z-graded rings is given. It shows, in particular, that a strongly Z-graded ring is left quasi-duo if and only if it is right quasi-duo. This gives a partial answer to a problem posed by Dugas and Lam…

In this paper we show if R is a filtered ring then we can define a quasi valuation. And if R is some kind of filtered ring then we can define a valuation. Then we prove some properties and relations for R.

The aim of this paper is to develop the theory of skew Armendariz and quasi-Armendariz modules over skew PBW extensions. We generalize the results of several works in the literature concerning Ore extensions to another non-commutative rings…

In this paper, we introduce a class of quasipolar rings which is a generalization of $J$-quasipolar rings. Let $R$ be a ring with identity. An element $a \in R$ is called {\it $\delta$-quasipolar} if there exists $p^2 = p\in comm^2(a)$ such…

Let $\mathbb{F}$ be a division ring. In this paper, we extent some of the main well-known results about the resultant of two univariate polynomials to the more general context of an Ore extension $\mathbb{F}[x;\sigma,\delta]$. Finally, some…

In this paper, we introduce a concept of weakly principally quasi-Baer *-rings in terms of central cover. We prove that a *-rings is a principally quasi-Baer *-rings if and only if it is weakly principally quasi-Baer *-rings with unity. A…