Narrow Results

Let $R$ be an arbitrary ring equipped with an automorphism $\alpha$ and an $\alpha$-derivation $\delta$. In this paper, we study on the relationship between the piecewise prime property of a ring $R$ and these of the skew inverse Laurent series ring $R((x^{-1};\alpha ,\delta ))$ and related rings. Furthermore, for a piecewise prime ring, we determine a large class of rings which have a generalized triangular matrix representation for which the diagonal rings are prime. Our results extend and unify many available results. Examples for illustration and delimitation of the theory are provided.

In this paper we characterize internally a TSA ring (i.e. a generalized triangular matrix ring with simple Artinian rings on the diagonal) in terms of its prime ideals. Also we show that the class of semiprimary quasi-Baer rings is a proper subclass of the class of TSA rings. Moreover, we generalize results of Harada, Small, and Teply on semiprimary rings. Finally we prove that under certain cardinality conditions a semiprimary quasi-Baer ring becomes semisimple Artinian.

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 a ring, $(S,\le )$ a strictly ordered monoid and $\omega :S\to \mathrm{\text{End}}(R)$ a monoid homomorphism. The skew generalized power series ring $R[[S,\omega ]]$ is a common generalization of (skew) polynomial rings, (skew) power series rings, (skew) Laurent polynomial rings, (skew) group rings, and Mal’cev–Neumann Laurent series rings. We initiate the study of the $(S,\omega )$-quasi-Armendariz condition on $R$, a generalization of the standard quasi-Armendariz condition from polynomials to skew generalized power series. The class of quasi-Armendariz rings includes semiprime rings, Armendariz rings, right (left) p.q.-Baer rings and right (left) PP rings. The $(S,\omega )$-quasi-Armendariz rings are closed under direct sums, upper triangular matrix rings, full matrix rings and Morita invariance. The $2\times 2$ formal upper triangular matrix rings of this class are characterized. We conclude some characterizations for a skew generalized power series ring to be semiprime, quasi-Baer, generalized quasi-Baer, primary, nilary, reflexive, ideal-symmetric and left AIP. Examples to illustrate and delimit the theory are provided.

Generalized matrix rings are ubiquitous in algebra and have relevant applications to analysis. A ring is quasi-Baer (respectively, right p.q.-Baer) in case the right annihilator of any ideal (respectively, principal ideal) is generated by an idempotent. A ring is called biregular if every principal ideal is generated by a central idempotent. In this paper, we identify the ideals and principal ideals, the annihilators of ideals, and the central and semi-central idempotents of a generalized $2\times 2$ matrix ring. We characterize the generalized matrix rings that are quasi-Baer, right p.q.-Baer, prime, and biregular. We provide examples to illustrate these concepts.

Let $R$ be a ring, $S$ a strictly ordered monoid and $\omega :S\to \mathrm{\text{End}}(R)$ a monoid homomorphism. In this paper, we investigate the problem when a skew generalized power series ring $R[[S,\omega ]]$ has the same triangulating dimension as the ring $R$. Furthermore, for a piecewise prime ring we determine a large class of the skew generalized power series ring which have a generalized triangular matrix representation for which the diagonal rings are prime. Our results extend and unify many existing results.

Let R be a ring with a derivation δ and R((x^-1; δ)) denote the pseudo-differential operator ring over R. We study the relations between the set of annihilators in R and the set of annihilators in R((x^-1; δ)). Among applications, it is shown that for an Armendariz ring R of pseudo-differential operator type, the ring R((x^-1; δ)) is Baer (resp., quasi-Baer, PP, right zip) if and only if R is a Baer (resp., quasi-Baer, PP, right zip) ring. For a δ-weakly rigid ring R, R((x^-1; δ)) is a left p.q.-Baer ring if and only if R is left p.q.-Baer and every countable subset of left semicentral idempotents of R has a generalized countable join in R.

Let R be a ring, (S, ≤) a strictly ordered monoid and ω : S → End(R) a monoid homomorphism. In this paper, we study the triangular matrix representation of skew generalized power series ring R[[S, ω]] which is a compact generalization of (skew) polynomial rings, (skew) power series rings, (skew) Laurent polynomials rings, (skew) Laurent power series rings, (skew) group rings, (skew) monoid rings, Mal'cev–Neumann rings and generalized power series rings. We investigate that if R is S-compatible and (S, ω)-Armendariz, then the skew generalized power series ring has same triangulating dimension as R. Furthermore, if R is a PWP ring, then skew generalized power series is also PWP ring.

For a ring endomorphism $\alpha$, a generalization of semiprime rings and right p.q.-Baer rings, which we call quasi-Armendariz rings of skew Hurwitz series type (or simply, $\alpha$-${𝒮}{\mathscr{H}}{𝒬}{𝒜}$), is introduced and studied. It is shown that the ${𝒮}{\mathscr{H}}{𝒬}{𝒜}$-rings are closed upper triangular matrix rings, full matrix rings and Morita invariance. Some characterizations for the skew Hurwitz series ring $(HR,\alpha )$ to be quasi-Baer, generalized quasi-Baer, primary, nilary, reflexive, ideal-symmetric and semiprime are concluded.