quasi-Baer Ring Research Articles

On Crossed Product Rings Over p.q.-Baer and Quasi-Baer Rings

In this paper, we consider a ring R and a monoid M equipped with a twisting map f: M×M -> U(R) and an action map ω: M -> Aut(R). The main objective of our study is to investigate the conditions under which the crossed product structure R⋊M is p.q.-Baer and quasi-Baer rings, and how this property relates to the p.q.-Baer property of R and the existence of a generalized join in I(R) for M-indexed subsets, where I(R) denotes the set of ideals of R. Additionally, we prove a connection between R being a left p.q.-Baer ring and the CM-quasi-Armendariz property. Moreover, we prove that for any element φ2=φ, there exist an idempotent element e2=e such that φ = ce. We then prove that R is quasi-Baer if and only if the crossed product structure R⋊M is quasi-Baer. Finally, we present novel results regarding various constructions for crossed products.

Rings which are Baer or quasi-Baer modulo a radical

Baer and quasi-Baer rings are important classes of algebraic objects, and their properties have roots in analysis. In this paper, we investigate rings R such that is Baer or quasi-Baer, where is either the Jacobson radical or the prime radical of R. Preliminary characterizations and results are obtained; in particular, we show that the property of being quasi-Baer is a Morita invariant.

Skew inverse Laurent series extensions of weakly principally quasi Baer rings

A ring [Formula: see text] is called weakly principally quasi Baer or simply (weakly p.q.-Baer) if the right annihilator of a principal right ideal is left [Formula: see text]-unital by left semicentral idempotents, which implies that [Formula: see text] modulo the right annihilator of any principal right ideal is flat. We study the relationship between the weakly p.q.-Baer property of a ring [Formula: see text] and those of the skew inverse series rings [Formula: see text] and [Formula: see text], for any automorphism [Formula: see text] and [Formula: see text]-derivation [Formula: see text] of [Formula: see text]. Examples to illustrate and delimit the theory are provided.

Differential Extensions of Weakly Principally Quasi-Baer Rings

A ring R is called weakly principally quasi-Baer or simply (weakly p.q.-Baer) if the right annihilator of a principal right ideal is right s-unital by right semicentral idempotents, which implies that R modulo, the right annihilator of any principal right ideal, is flat. We study the relationship between the weakly p.q.-Baer property of a ring R and those of the differential polynomial extension R[x;δ], the pseudo-differential operator ring R((x− 1;δ)), and also the differential inverse power series extension R[[x− 1;δ]] for any derivation δ of R. Examples to illustrate and delimit the theory are provided.

Semiprimeness, quasi-Baerness and prime radical of skew generalized power series rings

ABSTRACTLet R be a ring, (S,≤) a strictly ordered monoid and ω:S→End(R) a monoid homomorphism. The skew generalized power series ring R[[S,ω]] 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. In this paper we obtain necessary and sufficient conditions for the skew generalized power series ring R[[S,ω]] to be a semiprime, prime, quasi-Baer, or Baer ring. Furthermore, we study the prime radical of a skew generalized power series ring R[[S,ω]]. Our results extend and unify many existing results. In particular, we obtain new theorems on (skew) group rings, Mal’cev-Neumann Laurent series rings and the ring of generalized power series.

Direct sums of quasi-Baer modules

While the research work on quasi-Baer rings has been quite extensive in existing literature, the study of quasi-Baer modules introduced in 2004 remains quite limited with only little work available on the notion. In this paper, we extend the study of quasi-Baer modules. A characterization of a quasi-Baer module in terms of its endomorphism ring is obtained. We provide a complete characterization of arbitrary direct sums of quasi-Baer modules to be quasi-Baer. We also show characterizations of some important properties of quasi-Baer modules. Examples which delineate the concepts and the results are provided.

Annihilator Conditions related to the quasi-Baer Condition

We call a ring R an EGE-ring if for each $I \leq R$, which is generated by a subset of right semicentral idempotents there exists an idempotent $e$ such that $r(I) = eR$. The class EGE includes quasi-Baer, semiperfect rings (hence all local rings) and rings with a complete set of orthogonal primitive idempotents (hence all Noetherian rings) and is closed under direct product, full and upper triangular matrix rings, polynomial extensions (including formal power series, Laurent polynomials, and Laurent series) and is Morita invariant. Also we call $R$ an AE-ring if for each $I \unlhd R$, there exists a subset $S \subseteq S_{r}(R)$ such that $r(I) = r(RSR)$. The class AE includes the principally quasi-Baer ring and is closed under direct products, full and upper triangular matrix rings and is Morita invariant. For a semiprime ring $R$, it is shown that $R$ is an EGE (resp., AE)-ring if and only if the closure of any union of clopen subsets of $Spec(R)$ is open (resp., $Spec(R)$ is an EZ-space).

Endo-principally quasi-Baer modules

Analogous to left p.q.-Baer property of a ring [G. F. Birkenmeier, J. Y. Kim and J. K. Park, Principally quasi-Baer rings, Comm. Algebra29 (2001) 639–660], we say a right R-module M is endo-principallyquasi-Baer (or simply, endo-p.q.-Baer) if for every m ∈ M, lS(Sm) = Se for some e2= e ∈ S = EndR(M). It is shown that every direct summand of an endo-p.q.-Baer module inherits the property that any projective (free) module over a left p.q.-Baer ring is an endo-p.q.-Baer module. In particular, the endomorphism ring of every infinitely generated free right R-module is a left (or right) p.q.-Baer ring if and only if R is quasi-Baer. Furthermore, every principally right ℱℐ-extending right ℱℐ-𝒦-nonsingular ring is left p.q.-Baer and every left p.q.-Baer right ℱℐ-𝒦-cononsingular ring is principally right ℱℐ-extending.

Weakly principally quasi-Baer rings

We say a ring R with unity is weakly principally quasi-Baer or simply (weakly p.q.-Baer) if the right annihilator of a principal right ideal is right s-unital by right semicentral idempotents, which implies that R modulo the right annihilator of any principal right ideal is flat. It is proven that the weakly p.q.-Baer property is inherited by polynomial extensions and includes both left p.q.-Baer rings and right p.q.-Baer rings and is closed under direct products and Morita invariance. A ring R is weakly p.q.-Baer if and only if the upper triangular matrix ring Tn(R) is weakly p.q.-Baer. We extend a theorem of Kist for commutative Baer rings to weakly p.q.-Baer rings for which every prime ideal contains a unique minimal prime ideal without using topological arguments. It is also shown that there is an important subclass of weakly p.q.-Baer rings which have a nontrivial subdirect product representation.

When is a Sum of Annihilator Ideals an Annihilator Ideal?

We call a ring R a right SA-ring if for any ideals I and J of R there is an ideal K of R such that r(I) + r(J) = r(K). This class of rings is exactly the class of rings for which the lattice of right annihilator ideals is a sublattice of the lattice of ideals. The class of right SA-rings includes all quasi-Baer (hence all Baer) rings and all right IN-rings (hence all right selfinjective rings). This class is closed under direct products, full and upper triangular matrix rings, certain polynomial rings, and two-sided rings of quotients. The right SA-ring property is a Morita invariant. For a semiprime ring R, it is shown that R is a right SA-ring if and only if R is a quasi-Baer ring if and only if r(I) + r(J) = r(I ∩ J) for all ideals I and J of R if and only if Spec(R) is extremally disconnected. Examples are provided to illustrate and delimit our results.

A note on generalized quasi-Baer rings

A ring with identity is generalized quasi-Baer if for any ideal I of R, the right annihilator of In is generated by an idempotent for some positive integer n, depending on I. We study the generalized quasi-Baerness of R[x; σ; б] over a generalized quasi-Baer ring R where a is an automorphism of R.

Generalized APP-Rings

We say a ring R is (centrally) generalized left annihilator of principal ideal is pure (APP) if the left annihilator ℓ R (Ra) n is (centrally) right s-unital for every element a ∈ R and some positive integer n. The class of generalized left APP-rings includes generalized left (principally) quasi-Baer rings and left APP-rings (and hence left p.q.-Baer rings, right p.q.-Baer rings, and right PP-rings). The class of centrally generalized left APP-rings is closed under finite direct products, full matrix rings, and Morita invariance. The behavior of the (centrally) generalized left APP condition is investigated with respect to various constructions and extensions, and it is used to generalize many results on generalized PP-rings with IFP and semiprime left APP-rings. Moreover, we extend a theorem of Kist for commutative PP rings to centrally generalized left APP rings for which every prime ideal contains a unique minimal prime ideal without using topological arguments. Furthermore, we give a complete characterization of a considerably large family of centrally generalized left APP rings which have a sheaf representation.

Reflexive Property of Rings

Mason introduced the reflexive property for ideals, and then this concept was generalized by Kim and Baik, defining idempotent reflexive right ideals and rings. In this article, we characterize aspects of the reflexive and one-sided idempotent reflexive properties, showing that the concept of idempotent reflexive ring is not left-right symmetric. It is proved that a (right idempotent) reflexive ring which is not semiprime (resp., reflexive), can always be constructed from any semiprime (resp., reflexive) ring. It is also proved that the reflexive condition is Morita invariant and that the right quotient ring of a reflexive ring is reflexive. It is shown that both the polynomial ring and the power series ring over a reflexive ring are idempotent reflexive. We obtain additionally that the semiprimeness, reflexive property and one-sided idempotent reflexive property of a ring coincide for right principally quasi-Baer rings.

THE FACTOR RING OF A QUASI-BAER RING BY ITS PRIME RADICAL

The quasi-Baer condition of R/P(R) is investigated when R is a quasi-Baer ring, where P(R) is the prime radical of R. We provide an example of quasi-Baer ring R such that R/P(R) is not quasi-Baer. However, when P(R) is nilpotent, we prove that if R is a quasi-Baer (resp., Baer) ring, then R/P(R) is quasi-Baer (resp., Baer). Examples which illustrate and delimit the results of this paper are provided.

Polynomial extensions of generalized quasi-Baer rings

In this paper, we consider the behavior of polynomial rings over generalized quasi-Baer rings and show that the generalized quasi-Baer condition on a ring R is preserved by many polynomial extensions.

On Skew Quasi-Baer Rings

A ring R with an automorphism α and an α-derivation δ is called (α,δ)-quasi-Baer (resp., quasi-Baer) if the right annihilator of every (α,δ)-ideal (resp. ideal) of R is generated by an idempotent, as a right ideal. We show the left-right symmetry of (α, δ)-quasi Baer condition and prove that a ring R is (α, δ)-quasi Baer if and only if R[x; α, δ] is α-quasi Baer if and only if R[x; α, δ] is -quasi Baer for every extended derivation of δ. When R is a ring with IFP, then R is (α, δ)-Baer if and only if R[x; α, δ] is α-Baer if and only if R[x; α, δ] is -Baer for every extended α-derivation on R[x; α, δ] of δ. A rich source of examples for (α, δ)-quasi Baer rings is provided.

ON QUASI-RIGID IDEALS AND RINGS

Let <TEX>$\sigma$</TEX> be an endomorphism and I a <TEX>$\sigma$</TEX>-ideal of a ring R. Pearson and Stephenson called I a <TEX>$\sigma$</TEX>-semiprime ideal if whenever A is an ideal of R and m is an integer such that <TEX>$A{\sigma}^t(A)\;{\subseteq}\;I$</TEX> for all <TEX>$t\;{\geq}\;m$</TEX>, then <TEX>$A\;{\subseteq}\;I$</TEX>, where <TEX>$\sigma$</TEX> is an automorphism, and Hong et al. called I a <TEX>$\sigma$</TEX>-rigid ideal if <TEX>$a{\sigma}(a)\;{\in}\;I$</TEX> implies a <TEX>$a\;{\in}\;I$</TEX> for <TEX>$a\;{\in}\;R$</TEX>. Notice that R is called a <TEX>$\sigma$</TEX>-semiprime ring (resp., a <TEX>$\sigma$</TEX>-rigid ring) if the zero ideal of R is a <TEX>$\sigma$</TEX>-semiprime ideal (resp., a <TEX>$\sigma$</TEX>-rigid ideal). Every <TEX>$\sigma$</TEX>-rigid ideal is a <TEX>$\sigma$</TEX>-semiprime ideal for an automorphism <TEX>$\sigma$</TEX>, but the converse does not hold, in general. We, in this paper, introduce the quasi <TEX>$\sigma$</TEX>-rigidness of ideals and rings for an automorphism <TEX>$\sigma$</TEX> which is in between the <TEX>$\sigma$</TEX>-rigidness and the <TEX>$\sigma$</TEX>-semiprimeness, and study their related properties. A number of connections between the quasi <TEX>$\sigma$</TEX>-rigidness of a ring R and one of the Ore extension <TEX>$R[x;\;{\sigma},\;{\delta}]$</TEX> of R are also investigated. In particular, R is a (principally) quasi-Baer ring if and only if <TEX>$R[x;\;{\sigma},\;{\delta}]$</TEX> is a (principally) quasi-Baer ring, when R is a quasi <TEX>$\sigma$</TEX>-rigid ring.

Group Actions on Quasi-Baer Rings

AbstractA ring R is called quasi-Baer if the right annihilator of every right ideal of R is generated by an idempotent as a right ideal. We investigate the quasi-Baer property of skew group rings and fixed rings under a finite group action on a semiprime ring and their applications to C*-algebras. Various examples to illustrate and delimit our results are provided.

Ore Extensions of Quasi-Baer Rings

We first study the quasi-Baerness of R[x; σ, δ] over a quasi-Baer ring R when σ is an automorphism of R, obtaining an affirmative result. We next show that if R is a right principally quasi-Baer ring and σ is an automorphism of R with σ(e) = e for any left semicentral idempotent e ∈ R, then R[x; σ, δ] is right principally quasi-Baer. As a corollary, we have that R[x; δ] over a right principally quasi-Baer ring R is right principally quasi-Baer. Finally, we give conditions under which the quasi-Baernesses (right principal quasi-Baernesses) of R and R[x; σ, δ] are equivalent.

N-extended quasi-Baer rings

A ring with unit is said to be an n-extended right (principally) quasi-Baer ring if, for any proper (principal) right ideals I 1, …, I n , where n ≥ 2, the right annihilator of the product I 1 … I n is generated by an idempotent. A ring with unit is said to be an n-extended right (left) PP ring if the right (left, respectively) annihilator of the product x 1 … x n , where n ≥ 2, is generated by an idempotent for any nonidentity elements x 1, …, x n . The behavior of n-extended right (principally) quasi-Baer rings and right PP rings under various constructions and extensions is studied. These classes of rings are closed with respect to direct products and Morita equivalences. Examples illustrating the theory and outlining its frontiers are presented.