• East Asian mathematical journal
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• v.25 no.4
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• pp.433-440
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• 2009

Let R be a ring with unity, X the set of all nonzero, nonunits of R and G the group of all units of R. We will consider some group actions on X by G, the left (resp. right) regular action and the conjugate action. In this paper, by investigating these group actions we can have some results as follows: First, if E(R), the set of all nonzero nonunit idempotents of a unit-regular ring R, is commuting, then $o_{\ell}(x)\;=\;o_r(x)$, $o_c(x)\;=\;\{x\}$ for all $x\;{\in}\;X$ where $o_{\ell}(x)$ (resp. $o_r(x)$, $o_c(x)$) is the orbit of x under the left regular (resp. right regular, conjugate) action on X by G and R is abelian regular. Secondly, if R is a unit-regular ring with unity 1 such that G is a cyclic group and $2\;=\;1\;+\;1\;{\in}\;G$, then G is a finite group. Finally, if R is an abelian regular ring such that G is an abelian group, then R is a commutative ring.

• Bulletin of the Korean Mathematical Society
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• v.55 no.1
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• pp.25-40
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• 2018

We focus on the structure of the set of noncentral idempotents whose role is similar to one of central idempotents. We introduce the concept of quasi-Abelian rings which unit-regular rings satisfy. We first observe that the class of quasi-Abelian rings is seated between Abelian and direct finiteness. It is proved that a regular ring is directly finite if and only if it is quasi-Abelian. It is also shown that quasi-Abelian property is not left-right symmetric, but left-right symmetric when a given ring has an involution. Quasi-Abelian property is shown to do not pass to polynomial rings, comparing with Abelian property passing to polynomial rings.

• East Asian mathematical journal
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• v.36 no.1
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• pp.73-79
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• 2020

A ring R is called generalized regular if for every nonzero x in R there exists y in R such that xy is a nonzero idempotent. In this paper, we observe some equivalent conditions for the generalized regular rings that are abelian in terms of idempotents, and we also investigate the primitivity of an idempotent for such a ring. By using the investigation, we characterize such a kind of rings that are noetherian by showing that an abelian generalized regular ring R is noetherian if and only if R is isomorphic to a direct product of finitely many division rings. We also observe some interesting consequences of our results.

• Bulletin of the Korean Mathematical Society
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• v.59 no.4
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• pp.853-868
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• 2022

We study the structure of right regular commutators, and call a ring R strongly C-regular if ab - ba ∈ (ab - ba)²R for any a, b ∈ R. We first prove that a noncommutative strongly C-regular domain is a division algebra generated by all commutators; and that a ring (possibly without identity) is strongly C-regular if and only if it is Abelian C-regular (from which we infer that strong C-regularity is left-right symmetric). It is proved that for a strongly C-regular ring R, (i) if R/W(R) is commutative, then R is commutative; and (ii) every prime factor ring of R is either a commutative domain or a noncommutative division ring, where W(R) is the Wedderburn radical of R.

• Bulletin of the Korean Mathematical Society
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• v.42 no.4
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• pp.807-815
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• 2005

Let R be a ring with identity, X the set of all nonzero, nonunits of Rand G the group of all units of R. We will consider two group actions on X by G, the regular action and the conjugate action. In this paper, by investigating two group actions we can have some results as follows: First, if G is a finitely generated abelian group, then the orbit O(x) under the regular action on X by G is finite for all nilpotents x $\in$ X. Secondly, if F is a field in which 2 is a unit and F $\backslash\;\{0\}$ is a finitley generated abelian group, then F is finite. Finally, if G in a unit-regular ring R is a torsion group and 2 is a unit in R, then the conjugate action on X by G is trivial if and only if G is abelian if and only if R is commutative.

• Bulletin of the Korean Mathematical Society
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• v.53 no.3
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• pp.925-942
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• 2016

We study the structure of rings whose principal right ideals contain a sort of two-sided ideals, introducing right ${\pi}$-duo as a generalization of (weakly) right duo rings. Abelian ${\pi}$-regular rings are ${\pi}$-duo, which is compared with the fact that Abelian regular rings are duo. For a right ${\pi}$-duo ring R, it is shown that every prime ideal of R is maximal if and only if R is a (strongly) ${\pi}$-regular ring with $J(R)=N_*(R)$. This result may be helpful to develop several well-known results related to pm rings (i.e., rings whose prime ideals are maximal). We also extend the right ${\pi}$-duo property to several kinds of ring which have roles in ring theory.

• Korean Journal of Mathematics
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• v.28 no.1
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• pp.65-73
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• 2020

This article concerns a property of local rings and domains. A ring R is called weakly local if for every a ∈ R, a is regular or 1-a is regular, where a regular element means a non-zero-divisor. We study the structure of weakly local rings in relation to several kinds of factor rings and ring extensions that play roles in ring theory. We prove that the characteristic of a weakly local ring is either zero or a power of a prime number. It is also shown that the weakly local property can go up to polynomial (power series) rings and a kind of Abelian matrix rings.

• Journal of the Korean Mathematical Society
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• v.51 no.4
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• pp.751-771
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• 2014

We study the structure of idempotents in polynomial rings, power series rings, concentrating in the case of rings without identity. In the procedure we introduce right Insertion-of-Idempotents-Property (simply, right IIP) and right Idempotent-Reversible (simply, right IR) as generalizations of Abelian rings. It is proved that these two ring properties pass to power series rings and polynomial rings. It is also shown that ${\pi}$-regular rings are strongly ${\pi}$-regular when they are right IIP or right IR. Next the noncommutative right IR rings, right IIP rings, and Abelian rings of minimal order are completely determined up to isomorphism. These results lead to methods to construct such kinds of noncommutative rings appropriate for the situations occurred naturally in studying standard ring theoretic properties.

• Communications of the Korean Mathematical Society
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• v.11 no.2
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• pp.311-314
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• 1996

Let G be an abelian group of finite rink and E be the endomorphism ring of G. Then G is a left E-module by defining $f\cdota = f(a)$ for $f \in E$ and $a \in G$. In this case a condition for an E-module G to be regular is given.

• Journal of the Korean Mathematical Society
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• v.53 no.1
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• pp.217-232
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• 2016

We make a study of two generalizations of regular rings, concentrating our attention on the structure of idempotents. A ring R is said to be right attaching-idempotent if for $a{\in}R$ there exists $0{\neq}b{\in}R$ such that ab is an idempotent. Next R is said to be generalized regular if for $0{\neq}a{\in}R$ there exist nonzero $b{\in}R$ such that ab is a nonzero idempotent. It is first checked that generalized regular is left-right symmetric but right attaching-idempotent is not. The generalized regularity is shown to be a Morita invariant property. More structural properties of these two concepts are also investigated.