• 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.

• 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.

• 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 ${\sigma}$.

• Bulletin of the Korean Mathematical Society
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• v.47 no.2
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• pp.385-399
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• 2010

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

• 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.