• 대한수학회지
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• 제42권1호
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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 ${\sigma}$.

• 대한수학회보
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• 제41권4호
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• pp.657-664
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• 2004

We investigate skew power series of $\alpha$-rigid p.p.-rings, where $\alpha$ is an endomorphism of a ring R which is not assumed to be surjective. For an $\alpha$-rigid ring R, R[[${\chi};{\alpha}$]] is right p.p., if and only if R[[${\chi},{\chi}^{-1};{\alpha}$]] is right p.p., if and only if R is right p.p. and any countable family of idempotents in R has a join in I(R).

• 대한수학회보
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• 제49권4호
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• pp.715-736
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• 2012

Brown provided a structure theorem for a class of perfect ideals of grade 3 with type ${\lambda}$ > 0. We introduced a skew-symmetrizable matrix to describe a structure theorem for complete intersections of grade 4 in a Noetherian local ring. We construct a class of perfect ideals I of grade 3 with type 2 defined by a certain skew-symmetrizable matrix. We present the Hilbert function of the standard $k$-algebras R/I, where R is the polynomial ring $R=k[v_0,v_1,{\ldots},v_m]$ over a field $k$ with indeterminates $v_i$ and deg $v_i=1$.

• 대한수학회논문집
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• 제13권3호
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• pp.461-464
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• 1998

We generalize the Artin-Rees Theorem about the AR-property and study linking diagram in the set of prime ideals of the coordinate ring of quantum affine space.

• 충청수학회지
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• 제32권4호
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• pp.401-408
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• 2019

A ring R is called right (resp., left) nilpotent-duo if N(R)a ⊆ aN(R) (resp., aN(R) ⊆ N(R)a) for every a ∈ R, where N(R) is the set of all nilpotents in R. In this article we provide other proofs of known results and other computations for known examples in relation with right nilpotent-duo property. Furthermore we show that the left nilpotent-duo property does not go up to a kind of matrix ring.

• 대한수학회논문집
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• 제20권1호
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• pp.23-34
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• 2005

Let R be a ring with left identity. Let G : $R{\times}R{\to}R$ be a symmetric biadditive mapping and g the trace of G. Let ${\alpha}\;:\;R{\to}R$ be an endomorphism and ${\beta}\;:\;R{\to}R$ an epimorphism. In this paper we show the following: (i) Let R be 2-torsion-free. If g is (${\alpha},{\beta}$)-skew-commuting on R, then we have G = 0. (ii) If g is (${\beta},{\beta}$)-skew-centralizing on R, then g is (${\beta},{\beta}$)-commuting on R. (iii) Let $n{\ge}2$. Let R be (n+1)!-torsion-free. If g is n-(${\alpha},{\beta}$)-skew-commuting on R, then we have G = 0. (iv) Let R be 6-torsion-free. If g is 2-(${\alpha},{\beta}$)-commuting on R, then g is (${\alpha},{\beta}$)-commuting on R.

• 대한수학회보
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• 제58권4호
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• pp.865-876
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• 2021

Let 𝔽 be a commutative ring. A restricted skew polynomial extension over 𝔽 is a class of iterated skew polynomial 𝔽-algebras which include well-known quantized algebras such as the quantum algebra U_q(𝔰𝔩₂), Weyl algebra, etc. Here we obtain a necessary and sufficient condition in order to be restricted skew polynomial extensions over 𝔽. We also introduce a restricted Poisson polynomial extension which is a class of iterated Poisson polynomial algebras and observe that a restricted Poisson polynomial extension appears as semiclassical limits of restricted skew polynomial extensions. Moreover, we obtain usual as well as unusual quantized algebras of the same Poisson algebra as applications.

• 대한수학회보
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• 제49권5호
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• pp.1067-1079
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• 2012

Let M be a right R-module, (S, ${\leq}$) a strictly totally ordered monoid which is also artinian and ${\omega}:S{\rightarrow}Aut(R)$ a monoid homomorphism, and let $[M^{S,{\leq}}]_{[[R^{S,{\leq}},{\omega}]]$ denote the generalized inverse polynomial module over the skew generalized power series ring [[$R^{S,{\leq}},{\omega}$]]. In this paper, we prove that $[M^{S,{\leq}}]_{[[R^{S,{\leq}},{\omega}]]$ has the same uniform dimension as its coefficient module $M_R$, and that if, in addition, R is a right perfect ring and S is a chain monoid, then $[M^{S,{\leq}}]_{[[R^{S,{\leq}},{\omega}]]$ has the same couniform dimension as its coefficient module $M_R$.

• Korean Journal of Mathematics
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• 제21권1호
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• pp.41-53
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• 2013

Antoine studied conditions which are connected to the question of Amitsur of whether or not a polynomial ring over a nil ring is nil, introducing the notion of nil-Armendariz rings. Hizem extended the nil-Armendariz property for polynomial rings onto power-series rings, say nil power-serieswise rings. In this paper, we introduce the notion of power-serieswise CN rings that is a generalization of nil power-serieswise Armendariz rings. Finally, we study the nil-Armendariz property for Ore extensions and skew power series rings.

• 대한수학회지
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• 제55권5호
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• pp.1193-1205
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• 2018

We focus our attention on a ring property that nilpotents are contained in the Jacobson radical. This property is satisfied by NI and left (right) quasi-duo rings. A ring is said to be NJ if it satisfies such property. We prove the following: (i) $K{\ddot{o}}the^{\prime}s$ conjecture holds if and only if the polynomial ring over an NI ring is NJ; (ii) If R is an NJ ring, then R is exchange if and only if it is clean; and (iii) A ring R is NJ if and only if so is every (one-sided) corner ring of R.