• 대한수학회논문집
• /
• 제32권2호
• /
• pp.267-276
• /
• 2017

In this paper we introduce rings that satisfy regular 1-stable range. These rings are left-right symmetric and are generalizations of unit 1-stable range. We investigate characterizations of these kind of rings and show that these rings are closed under matrix rings and Morita Context rings.

• 대한수학회논문집
• /
• 제33권4호
• /
• pp.1075-1082
• /
• 2018

Let R be a commutative ring, G be an Abelian group, and let RG be the group ring. We say that RG is a U-group ring if a is a unit in RG if and only if ${\epsilon}(a)$ is a unit in R. We show that RG is a U-group ring if and only if G is a p-group and $p{\in}J(R)$. We give some properties of U-group rings and investigate some properties of well known rings, such as Hermite rings and rings with stable range, in the presence of U-group rings.

• 대한수학회보
• /
• 제39권2호
• /
• pp.317-326
• /
• 2002

In this paper, we establish necessary and sufficient conditions for an exchange ring R to satisfy the n-stable range condition. It is shown that an exchange ring R satisfies the n-stable range condition if and only if for any regular a $\in$ R$^n$, there exists a unimodular u $\in$$^n$ R such that au $\in$ R is a group member, and if and only if whenever a$\simeq$$_n$b with a $\in$ R, b $\in$ M$_n$(R), there exist u $\in$ R$^n$, v $\in$$^n$ R such that a = ubv with uv = 1. As an application, we observe that exchange rings satisfying the n-stable range condition can be characterized by Drazin inverses. These also give nontrivial generalizations of [7, Theorem 10], [13, Theorem 10], [15, Theorem] and [16, Theorem. 2A].

• 대한수학회지
• /
• 제52권4호
• /
• pp.839-851
• /
• 2015

A *-ring R is called (strongly) *-clean if every element of R is the sum of a projection and a unit (which commute with each other). In this note, some properties of *-clean rings are considered. In particular, a new class of *-clean rings which called strongly ${\pi}$-*-regular are introduced. It is shown that R is strongly ${\pi}$-*-regular if and only if R is ${\pi}$-regular and every idempotent of R is a projection if and only if R/J(R) is strongly regular with J(R) nil, and every idempotent of R/J(R) is lifted to a central projection of R. In addition, the stable range conditions of *-clean rings are discussed, and equivalent conditions among *-rings related to *-cleanness are obtained.

• 대한수학회지
• /
• 제45권3호
• /
• pp.741-756
• /
• 2008

In this paper, we introduce a new class of rings, SB-rings. We establish various properties of this concept. These shows that, in several respects, SB-rings behave like rings satisfying unit 1-stable range. We will give necessary and sufficient conditions under which a semilocal ring is a SB-ring. Furthermore, we extend these results to exchange rings with all primitive factors artinian. For such rings, we observe that the concept of the SB-ring coincides with Goodearl-Menal condition. These also generalize the results of Huh et al., Yu and the author on rings generated by their units.

• 대한수학회지
• /
• 제57권3호
• /
• pp.793-807
• /
• 2020

Stable range of rings is a unifying concept for problems related to the substitution and cancellation of modules. The newly appeared element-wise setting for the simplest case of stable range one is tempting to study the lifting property modulo ideals. We study the lifting of elements having (idempotent) stable range one from a quotient of a ring R modulo a two-sided ideal I by providing several examples and investigating the relations with other lifting properties, including lifting idempotents, lifting units, and lifting of von Neumann regular elements. In the case where the ring R is a left or a right duo ring, we show that stable range one elements lift modulo every two-sided ideal if and only if R is a ring with stable range one. Under a mild assumption, we further prove that the lifting of elements having idempotent stable range one implies the lifting of von Neumann regular elements.

• Kyungpook Mathematical Journal
• /
• 제59권4호
• /
• pp.603-615
• /
• 2019

Let R be an associative unital ring with an endomorphism α and α-derivation δ. The constant products of elements in Ore extension rings, when the coefficient ring is reversible, is investigated. We show that if f(x) = ∑ⁿ_i=0 a_ixⁱ and g(x) = ∑^m_j=0 b_jx^j be nonzero elements in Ore extension ring R[x; α, δ] such that g(x)f(x) = c ∈ R, then there exist non-zero elements r, a ∈ R such that rf(x) = ac, when R is an (α, δ)-compatible ring which is reversible. Among applications, we give an exact characterization of the unit elements in R[x; α, δ], when the coeficient ring R is (α, δ)-compatible. Furthermore, it is shown that if R is a weakly 2-primal ring which is (α, δ)-compatible, then J(R[x; α, δ]) = N iℓ(R)[x; α, δ]. Some other applications and examples of rings with this property are given, with an emphasis on certain classes of NI rings. As a consequence we obtain generalizations of the many results in the literature. As the final part of the paper we construct examples of rings that explain the limitations of the results obtained and support our main results.

• 대한수학회보
• /
• 제41권4호
• /
• pp.599-608
• /
• 2004

Let R be an associative ring with identity and J(R) be the Jacobson radical of R. In this paper we investigate the generalization of the Jacobson radical of R, J＊ (R) say. Also we study the rings that J＊(R) = J(R).

• Macromolecular Research
• /
• 제13권1호
• /
• pp.14-18
• /
• 2005

Under irradiation in a microwave oven, six novel poly(amide-imide)s containing tetrahydropyrimidinone, tetrahydro-2-thioxopyrimidine moieties and trimellitic rings in their main chains were synthesized through the polycondensation reaction of N,N'-(4,4'-diphenylether)bis(trimellitimide) diacid chloride with six different derivatives of tetrahydropyrimidinone and tetrahydro-2-thioxopyrimidine in the presence of a small amount of a polar organic medium, such as o-cresol. The polycondensation proceeded rapidly and completed within 7-9 min, producing a series of new poly(amide-imide)s in high yield that showed inherent viscosities in the range 0.33-0.52 dL/g. These poly(amide-imide)s were characterized by elemental analysis, viscosity measurement, thermal gravimetric analysis, solubility test, and FT-IR spectroscopy. All of the polymers were soluble at room temperature in polar solvents such as N,N-dimethylacetamide, N,N-dimethylformamide, dimethylsulfoxide, tetrahydrofuran, and N-methyl-2-pyrrolidone.

• Korean Chemical Engineering Research
• /
• 제59권1호
• /
• pp.94-99
• /
• 2021

2단 Rushton turbine 날개를 장착한 접시바닥형 교반조 내의 유동 상태가 어떤 회전수에서 변화하는 현상을 찾아냈다. 층류 영역에서는 회전수를 변화시켜도 안정적인 도넛 링이 형성되어 혼합 패턴에 특이한 변동은 관찰되지 않았다. 전이영역에서의 평바닥형 교반조에서는 회전수에 변화를 주어도 두꺼운 2개의 미혼합 도넛 링에 변화가 없는 반면, 접시바닥형 교반조에서는 회전수가 450 rpm이 되면 두꺼운 2개의 미혼합 도넛 링이 아주 얇은 3개의 도넛 링으로 바뀌어 혼합이 개선되었다. 접시바닷형 교반조에서는 Re=138~178의 영역에서 링 모양의 미혼합 영역이 3곳에서 나타났고 그 크기도 컸지만, 평바닥형 교반조에서는 Re=116~176에서 링모양의 미혼합 영역은 2곳에서 나타났으며 그 크기도 작았다. 이러한 현상이 관찰되는 조건은 전이영역으로, 방해판을 부착하였을 때가 방해판이 없을 때와 비교하여 동력수가 커지기 시작하는 영역임을 알 수 있었다. 또한 교반 레이놀즈수가 300을 넘어 유동 상태에 약간의 난류가 섞이게 되면, 이 같은 흐름 양상의 단절이 해소되어 완전히 혼합되었다.