• 대한수학회지
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• 제52권6호
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• pp.1161-1178
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• 2015

In this paper, we investigate the insertion-of-factors-property (simply, IFP) on skew polynomial rings, introducing the concept of strongly ${\sigma}-IFP$ for a ring endomorphism ${\sigma}$. A ring R is said to have strongly ${\sigma}-IFP$ if the skew polynomial ring R[x;${\sigma}$] has IFP. We examine some characterizations and extensions of strongly ${\sigma}-IFP$ rings in relation with several ring theoretic properties which have important roles in ring theory. We also extend many of related basic results to the wider classes, and so several known results follow as consequences of our results.

• 대한수학회논문집
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• 제34권2호
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• pp.361-374
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• 2019

Let R be a ring, ($S,{\leq}$) a strictly ordered monoid, and ${\omega}:S{\rightarrow}End(R)$ a monoid homomorphism. In [18], Mazurek, and Ziembowski investigated when the skew generalized power series ring $R[[S,{\omega}]]$ is a domain satisfying the ascending chain condition on principal left (resp. right) ideals. Following [18], we obtain necessary and sufficient conditions on R, S and ${\omega}$ such that the skew generalized power series ring $R[[S,{\omega}]]$ is a right or left Archimedean domain. As particular cases of our general results we obtain new theorems on the ring of arithmetical functions and the ring of generalized power series. Our results extend and unify many existing results.

• 대한수학회보
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• 제45권2호
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• pp.285-297
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• 2008

For an endomorphism ${\alpha}$ of a ring R, the endomorphism ${\alpha}$ is called semicommutative if ab=0 implies $aR{\alpha}(b)$=0 for a ${\in}$ R. A ring R is called ${\alpha}$-semicommutative if there exists a semicommutative endomorphism ${\alpha}$ of R. In this paper, various results of semicommutative rings are extended to ${\alpha}$-semicommutative rings. In addition, we introduce the notion of an ${\alpha}$-skew power series Armendariz ring which is an extension of Armendariz property in a ring R by considering the polynomials in the skew power series ring $R[[x;\;{\alpha}]]$. We show that a number of interesting properties of a ring R transfer to its the skew power series ring $R[[x;\;{\alpha}]]$ and vice-versa such as the Baer property and the p.p.-property, when R is ${\alpha}$-skew power series Armendariz. Several known results relating to ${\alpha}$-rigid rings can be obtained as corollaries of our results.

• 대한수학회지
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• 제53권2호
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• pp.381-401
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• 2016

According to Jacobson [31], a right ideal is bounded if it contains a non-zero ideal, and Faith [15] called a ring strongly right bounded if every non-zero right ideal is bounded. From [30], a ring is strongly right AB if every non-zero right annihilator is bounded. In this paper, we introduce and investigate a particular class of McCoy rings which satisfy Property (A) and the conditions asked by Nielsen [42]. It is shown that for a u.p.-monoid M and ${\sigma}:M{\rightarrow}End(R)$ a compatible monoid homomorphism, if R is reversible, then the skew monoid ring R * M is strongly right AB. If R is a strongly right AB ring, M is a u.p.-monoid and ${\sigma}:M{\rightarrow}End(R)$ is a weakly rigid monoid homomorphism, then the skew monoid ring R * M has right Property (A).

• 대한수학회지
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• 제52권4호
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• pp.663-683
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• 2015

In this note we study the structures of power-serieswise Armendariz rings and IFP rings when they are skewed by ring endomor-phisms (or automorphisms). We call such rings skew power-serieswise Armendariz rings and skew IFP rings, respectively. We also investigate relationships among them and construct necessary examples in the process. The results argued in this note can be extended to the ordinary ring theoretic properties of power-serieswise Armendariz rings, IFP rings, and near-related rings.

• Kyungpook Mathematical Journal
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• 제57권3호
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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.

• 대한수학회논문집
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• 제21권1호
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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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• 제22권2호
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• pp.217-234
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• 2014

Mason extended the reflexive property for subgroups to right ideals, and examined various connections between these and related concepts. A ring was usually called reflexive if the zero ideal satisfies the reflexive property. We here study this property skewed by ring endomorphisms, introducing the concept of an ${\alpha}$-skew reflexive ring, where is an endomorphism of a given ring.

• 대한수학회논문집
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• 제39권1호
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• pp.1-10
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• 2024

In this paper we will demonstrate some results on a prime ring with involution by introducing two generalized derivations acting on symmetric and skew symmetric elements. This approach allows us to generalize some well known results. Furthermore, we provide examples to show that various restrictions imposed in the hypotheses of our theorems are not superfluous.

• 대한수학회지
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• 제42권2호
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• pp.353-363
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• 2005

For a ring endomorphism $\alpha$ and an $\alpha$-derivation $\delta$, we introduce ($\alpha$, $\delta$)-skew Armendariz rings which are a generalization of $\alpha$-rigid rings and Armendariz rings, and investigate their properties. A semi prime left Goldie ring is $\alpha$-weak Armendariz if and only if it is $\alpha$-rigid. Moreover, we study on the relationship between the Baerness and p.p. property of a ring R and these of the skew polynomial ring R[x; $\alpha$, $\delta$] in case R is ($\alpha$, $\delta$)-skew Armendariz. As a consequence we obtain a generalization of [11], [14] and [16].