• Communications of the Korean Mathematical Society
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• v.32 no.2
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• pp.267-276
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• 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.

• Communications of the Korean Mathematical Society
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• v.33 no.4
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• pp.1075-1082
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• 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.

• Bulletin of the Korean Mathematical Society
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• v.39 no.2
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• pp.317-326
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• 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].

• Journal of the Korean Mathematical Society
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• v.52 no.4
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• pp.839-851
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• 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.

• Journal of the Korean Mathematical Society
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• v.45 no.3
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• pp.741-756
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• 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.

• Journal of the Korean Mathematical Society
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• v.57 no.3
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• pp.793-807
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• 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
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• v.59 no.4
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• pp.603-615
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• 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.

• Bulletin of the Korean Mathematical Society
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• v.41 no.4
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• pp.599-608
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• 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
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• v.13 no.1
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• pp.14-18
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• 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
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• v.59 no.1
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• pp.94-99
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• 2021

It was found that mixing patterns suddenly changed at an impeller rotation speed in a dished bottom vessel with dual Rushton turbines. Two isolated mixing regions like doughnuts rings generated at a low rotational speed and three isolated mixing regions generated at a higher speed. This phenomenon was observed at the mixing condition in transition area, where the power number with baffle was the same as that without baffle. We found a phenomenon in which the flow state in a dish-bottom agitation tank equipped with a two-stage Rushton turbine blade changes at a certain rotational speed. In the laminar flow region, the isolated stable donut rings were formed even when the rotational speed was changed, and no specific variation in the mixing pattern was observed. In the transition region, the two isolated thick unmixed donut rings do not change even if the rotation speed is changed in the flat bottom vessel, whereas in the dished bottom vessel, when the rotation speed is 450 rpm, the two isolated thick unmixed donut rings are changed to three isolated thin donut rings and then improved mixing. In the dished bottom vessel, in the range of Re=138~178, the isolated ring-shaped unmixed region appeared in three places and the size was also large. But in the flat bottom vessel, the isolated thick ring-shaped unmixed region appeared in two places in Re=116~176 and the size was also small. It appeared in two places, and the size was also small. The condition in which this phenomenon is observed is a transition region, and it was found that when the baffle plate is attached, the power number starts to increase compared to when the baffle plate is not present. In addition, when the mixing Reynolds number exceeded 300 and a slight turbulence was mixed in the flow state, the disconnection of these flow pattern was resolved and the mixture was completely mixed.