• Communications of the Korean Mathematical Society
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• v.21 no.1
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• pp.1-9
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• 2006

A ring R is called (left principally) quasi-Baer if the left annihilator of every (principal) left ideal of R is generated by an idempotent. Let R be a ring, G be an ordered monoid acting on R by $\beta$ and R be G-compatible. It is shown that R is (left principally) quasi-Baer if and only if skew monoid ring $R_{\beta}[G]$ is (left principally) quasi-Baer. If G is an abelian monoid, then R is (left principally) quasi-Baer if and only if the Cohn-Jordan extension $A(R,\;\beta)$ is (left principally) quasi-Baer if and only if left Ore quotient ring $G^{-1}R_{\beta}[G]$ is (left principally) quasi-Baer.

• Korean Journal of Mathematics
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• v.17 no.3
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• pp.233-236
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• 2009

We investigate in this paper some equivalent conditions for right principally quasi-Baer rings to be reflexive. Using these results we are able to prove that if R is a reflexive right principally quasi-Baer ring then R is a left principally quasi-Baer ring. In addition, for an idempotent reflexive principally quasi-Baer ring R we show that R is prime if and only if R is torsion free.

• East Asian mathematical journal
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• v.17 no.2
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• pp.275-286
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• 2001

In this paper, a skew polynomial ring $R[x;\alpha]$ of a ring R with a monomorphism $\alpha$ are investigated as follows: For a reduced ring R, assume that $\alpha(P){\subseteq}P$ for any minimal prime ideal P in R. Then (i) $R[x;\alpha]$ is a reduced ring, (ii) a ring R is Baer(resp. quasi-Baer, p.q.-Baer, a p.p.-ring) if and only if the skew polynomial ring $R[x;\alpha]$ is Baer(resp. quasi-Baer, p.q.-Baer, a p.p.-ring).

• Bulletin of the Korean Mathematical Society
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• v.42 no.3
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• pp.527-534
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• 2005

We consider countable rings with ascending chain condition on right annihilators. We determine the structure of a countable right p-injective Baer ring, a countable semi prime quasi-Baer ring and a countable quasi-Baer biregular ring.

• Kyungpook Mathematical Journal
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• v.47 no.1
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• pp.21-30
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• 2007

We say a module $M_R$ a semicommutative module if for any $m{\in}M$ and any $a{\in}R$, $ma=0$ implies $mRa=0$. This paper gives various properties of reduced, Armendariz, Baer, Quasi-Baer, p.p. and p.q.-Baer rings to extend to modules. In addition we also prove, for a p.p.-ring R, R is semicommutative iff R is Armendariz. Let R be an abelian ring and $M_R$ be a p.p.-module, then $M_R$ is a semicommutative module iff $M_R$ is an Armendariz module. For any ring R, R is semicommutative iff A(R, ${\alpha}$) is semicommutative. Let R be a reduced ring, it is shown that for number $n{\geq}4$ and $k=[n=2]$, $T^k_n(R)$ is semicommutative ring but $T^{k-1}_n(R)$ is not.

• Kyungpook Mathematical Journal
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• v.51 no.4
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• pp.375-384
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• 2011

We study Baer and quasi-Baer modules over some classes of rings. We also introduce a new class of modules called AI-modules, in which the kernel of every nonzero endomorphism is contained in a proper direct summand. The main results obtained here are: (1) A module is Baer iff it is an AI-module and has SSIP. (2) For a perfect ring R, the direct sum of Baer modules is Baer iff R is primary decomposable. (3) Every injective R-module is quasi-Baer iff R is a QI-ring.

• Journal of the Korean Mathematical Society
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• v.42 no.1
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• pp.53-63
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• 2005

Let R be a ring R and ${\sigma}$ be an endomorphism of R. R is called ${\sigma}$-rigid (resp. reduced) if $a{\sigma}r(a) = 0 (resp{\cdot}a^2 = 0)$ for any $a{\in}R$ implies a = 0. An ideal I of R is called a ${\sigma}$-ideal if ${\sigma}(I){\subseteq}I$. R is called ${\sigma}$-quasi-Baer (resp. right (or left) ${\sigma}$-p.q.-Baer) if the right annihilator of every ${\sigma}$-ideal (resp. right (or left) principal ${\sigma}$-ideal) of R is generated by an idempotent of R. In this paper, a skew polynomial ring A = R[$x;{\sigma}$] of a ring R is investigated as follows: For a ${\sigma}$-rigid ring R, (1) R is ${\sigma}$-quasi-Baer if and only if A is quasi-Baer if and only if A is $\={\sigma}$-quasi-Baer for every extended endomorphism $\={\sigma}$ on A of ${\sigma}$ (2) R is right ${\sigma}$-p.q.-Baer if and only if R is ${\sigma}$-p.q.-Baer if and only if A is right p.q.-Baer if and only if A is p.q.-Baer if and only if A is $\={\sigma}$-p.q.-Baer if and only if A is right $\={\sigma}$-p.q.-Baer for every extended endomorphism $\={\sigma}$ on A of ${\sigma}$.

• Kyungpook Mathematical Journal
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• v.57 no.3
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• pp.401-417
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• 2017

Ever since their introduction, skew PBW ($Poincar{\acute{e}}$-Birkhoff-Witt) extensions of rings have kept growing in importance, as researchers characterized their properties (such as primeness, Krull and Goldie dimension, homological properties, etc.) in terms of intrinsic properties of the base ring, and studied their relations with other fields of mathematics, as for example quantum mechanics theory. Many rings and algebras arising in quantum mechanics can be interpreted as skew PBW extensions. Our aim in this paper is to study skew PBW extensions of Baer, quasi-Baer, principally projective and principally quasi-Baer rings, in the case when the base ring R is not assumed to be reduced. We just impose some mild compatibleness over the base ring R, and prove that these properties are stable over this kind of extensions.

• East Asian mathematical journal
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• v.38 no.1
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• pp.1-12
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• 2022

Let R be a ring with an endomorphism σ and a σ-derivation δ. R is called (σ, δ)-Baer (resp. (σ, δ)-quasi-Baer, (σ, δ)-p.q.-Baer, (σ, δ)-p.p.) if the right annihilator of every right (σ, δ)-set (resp., (σ, δ)-ideal, principal (σ, δ)-ideal, (σ, δ)-element) of R is generated by an idempotent of R. In this paper, for a given Ore extension A = R[x; σ, δ] of R, the following properties are investigated: If R is a σ-rigid ring in which σ and δ commute, then (1) R is (σ, δ)-Baer if and only if R is (σ, δ)-quasi-Baer if and only if A is (${\bar{\sigma}},\;{\bar{\delta}}$)-Baer if and only if A is (${\bar{\sigma}},\;{\bar{\delta}}$)-quasi-Baer; (2) R is (σ, δ)-p.p. if and only if R is (σ, δ)-p.q.-Baer if and only if A is (${\bar{\sigma}},\;{\bar{\delta}}$)-p.p. if and only if A is (${\bar{\sigma}},\;{\bar{\delta}}$)-p.q.-Baer.

• Journal of the Korean Mathematical Society
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• v.44 no.6
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• pp.1267-1279
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• 2007

For a ring endomorphism ${\alpha}$ and an ${\alpha}-derivation\;{\delta}$ of a ring R, we study relation between the set of annihilators in R and the set of annihilators in nearring $R[x;{\alpha},{\delta}]\;and\;R_0[[x;{\alpha}]]$. Also we extend results of Armendariz on the Baer and p.p. conditions in a polynomial ring to certain analogous annihilator conditions in a nearring of skew polynomials. These results are somewhat surprising since, in contrast to the skew polynomial ring and skew power series case, the nearring of skew polynomials and skew power series have substitution for its "multiplication" operation.