• Bulletin of the Korean Mathematical Society
• /
• v.46 no.6
• /
• pp.1091-1097
• /
• 2009

In this paper, we investigate the extensions of generalized stable rings. It is shown that a ring R is a generalized stable ring if and only if R has a complete orthogonal set {e$_1$, . . . , e$_n$} of idempotents such that e$_1$Re$_1$, . . . , e$_n$Re$_n$ are generalized stable rings. Also, we prove that a ring R is a generalized stable ring if and only if R[[X]] is a generalized stable ring if and only if T(R,M) is a generalized stable ring.

• Journal of the Korean Mathematical Society
• /
• v.57 no.3
• /
• 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.

• Journal of the Korean Mathematical Society
• /
• v.46 no.2
• /
• pp.257-269
• /
• 2009

A ring R is a weakly stable ring provided that aR + bR = R implies that there exists $y\;{\in}\;R$ such that $a\;+\;by\;{\in}\;R$ is right or left invertible. In this article, we characterize weakly stable rings by virtue of $2{\times}2$ invertible matrices over them. It is shown that a ring R is a weakly stable ring if and only if for any $A\;{\in}GL_2(R)$, there exist two invertible lower triangular L and K and an invertible upper triangular U such that A = LUK, where two of L, U and K have diagonal entries 1. Related results are also given. These extend the work of Nagarajan et al.

• Journal of the Korean Mathematical Society
• /
• v.44 no.4
• /
• pp.903-913
• /
• 2007

A ring R has weakly stable range one provided that aR+bR=R implies that there exists a $y{\in}R$ such that $a+by{\in}R$ is right or left invertible. We prove, in this paper, that every regular element in an exchange ring having weakly stable range one is the sum of an idempotent and a weak unit. This generalize the corresponding result of one-sided unit-regular ring. Extensions of power comparability and power cancellation are also studied.

• Bulletin of the Korean Mathematical Society
• /
• v.39 no.2
• /
• 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].

• Bulletin of the Korean Mathematical Society
• /
• v.53 no.1
• /
• pp.153-161
• /
• 2016

It is shown that the algebra $\mathfrak{H}^{\infty}$ of bounded Dirichlet series is not a coherent ring, and has infinite Bass stable rank. As corollaries of the latter result, it is derived that $\mathfrak{H}^{\infty}$ has infinite topological stable rank and infinite Krull dimension.

• Communications of the Korean Mathematical Society
• /
• v.33 no.4
• /
• 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.

• Journal of Advanced Marine Engineering and Technology
• /
• v.37 no.7
• /
• pp.752-759
• /
• 2013

This paper presents a method for analyzing and designing an extremely miniaturized planar ring hybrid, using the combination of parallel and diagonally shorted coupled lines. In contrast to conventional miniaturized coupled line filters, it is proven that the required electrical length of transmission line can be largely reduced to even a few degrees, not only effectively suppressing the spurious passband but also approximately maintaining the same characteristic around the stable center frequency. A ring hybrid filter at center frequency of 1 GHz was fabricated on the FR4 epoxy glass cloth copper-clad plat (CCL) PCB substrate. The insertion loss of a ring hybrid filter with the die area of $30mm{\times}30mm$ is -4.68 dB. Simulated results are well agreed with the measurements.

• Journal of the Korean Mathematical Society
• /
• v.45 no.3
• /
• 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.

• Journal of the Korean Mathematical Society
• /
• v.52 no.4
• /
• 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.