I-SEMIREGULAR RINGS

한준철,심효섭 영남수학회 2020 East Asian mathematical journal Vol.36 No.3

Let R be a ring with unity, and let I be an ideal of R. Then R is called I-semiregular if for every a ∈ R there exists b ∈ R such that ab is an idempotent of R and a − aba ∈ I. In this paper, basic properties of I-semiregularity are investigated, and some equivalent conditions to the primitivity of e are observed for an idempotent e of an I-semiregular ring R such that I ∩eR = (0). For an abelian regular ring R with the ascending chain condition on annihilators of idempotents of R, it is shown that R is isomorphic to a direct product of a finite number of division rings, as a consequence of the observations.

Group actions in a regular ring

한준철 대한수학회 2005 대한수학회보 Vol.42 No.4

Let R be a ring with identity, X the set of all nonzero,nonunits of R and G the group of all units of R. We will considertwo group actions on X by G, the regular action and the conjugateaction. In this paper, by investigating two group actions we canhave some results as follows: First, if G is a nitely generatedabelian group, then the orbit O(x) under the regular action on X by G is nite for all nilpotents x 2 X . Secondly, if F is a eld in which 2 is a unit and F-{0} is a finitley generated abelian group,then F is nite. Finally, ifG in a unit-regular ring R is a torsiongroup 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.

Skew polynomial rings over $\sigma$-quasi-Baer and $\sigma$-principally quasi-Baer rings

한준철 대한수학회 2005 대한수학회지 Vol.42 No.1

Let R be a ring R and ¾ be an endomorphism of R. R is called ¾-rigid (resp. reduced) if a¾(a) = 0 (resp. a2 = 0) for any a 2 R implies a = 0. An ideal I of R is called a ¾-ideal if ¾(I) µ I. R is called ¾-quasi-Baer (resp. right (or left) ¾-p:q:-Baer) if the right annihilator of every ¾-ideal (resp. right (or left) principal ¾-ideal) of R is generated by an idempotent of R. In this paper, a skew polynomial ring A = R[x; ¾] of a ring R is investigated as follows: For a ¾-rigid ring R, (1) R is ¾-quasi-Baer if and only if A is quasi-Baer if and only if A is ¹¾-quasi-Baer for every extended endomorphism ¹¾ on A of ¾; (2) R is right ¾-p.q.-Baer if and only if R is ¾-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 ¹¾-p.q.-Baer if and only if A is right ¹¾-p.q.-Baer for every extended endomorphism ¹¾ on A of ¾.

Ore Extensions over $\sigma$-rigid Rings

한준철,이양,심효섭 영남수학회 2022 East Asian mathematical journal Vol.38 No.1

Let $R$ be a ring with an endomorphism $\sigma$ and a $\sigma$-derivation$\delta$. $R$ is called ($\sigma$, $\delta$)-$Baer$(resp. ($\sigma$, $\delta$)-$quasi$-$Baer$, ($\sigma$,$\delta$)-$p.q.$-$Baer$, ($\sigma$, $\delta$)-$p.p.$) if the rightannihilator of every right ($\sigma$, $\delta$)-set (resp.,($\sigma$, $\delta$)-ideal, principal ($\sigma$, $\delta$)-ideal,($\sigma$, $\delta$)-element) of $R$ is generated by an idempotentof $R$. In this paper, for a given Ore extension $A$ = $R[x; \sigma, \delta]$of $R$, the following properties are investigated:If $R$ is a $\sigma$-rigid ringin which $\sigma$ and $\delta$ commute, then(1) $R$ is ($\sigma, \delta)$-Baer if and only if$R$ is ($\sigma, \delta)$-quasi-Baer if and only if $A$ is ($\bar{\sigma},\bar{\delta})$-Baer if and only if $A$ is ($\bar{\sigma},\bar{\delta})$-quasi-Baer;(2) $R$ is ($\sigma, \delta)$-p.p. if and only if$R$ is ($\sigma, \delta)$-$p.q.$-Baer if and only if $A$ is ($\bar{\sigma},\bar{\delta})$-$p.p.$ if and only if $A$ is ($\bar{\sigma},\bar{\delta})$-$p.q.$-Baer.

On idempotents in relation with regularity

한준철,이양,박상원,성효진,윤상조 대한수학회 2016 대한수학회지 Vol.53 No.1

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 {\it 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 {\it 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.

Prime Radicals of Skew Laurent Polynomial Rings

한준철 대한수학회 2005 대한수학회보 Vol.42 No.3

Let R be a ring withan automorphism sigma. An ideal I of R is sigma-idealof R if sigma(I) = I. A proper ideal P of R is{itsigma-prime ideal/} of R if P is a sigma-ideal of Rand for sigma-ideals I and J of R, IJ subseteq Pimplies that I subseteq P or J subseteq P. A properideal Q of R is {itsigma-semiprime ideal/} of Q if Q isa sigma-ideal and for a sigma-ideal I of R, I^{2}subseteq Q implies that I subseteq Q. Thesigma-prime radical is defined by the intersection of allsigma-prime ideals of R and is denoted by P_{sigma}(R).In this paper, the following results are obtained: (1)For aprincipal ideal domain R, P_{sigma}(R) is the smallestsigma-semiprime ideal of R; (2)For any ring R with anautomorphism sigma and for a skew Laurent polynomial ring R[x,x^{-1}; sigma], the prime radical of R[x, x^{-1}; sigma] isequal to P_{sigma}(R)[x, x^{-1}; sigma].

Unit-duo rings and related graphs of zero divisors

한준철,이양,박상원 대한수학회 2016 대한수학회보 Vol.53 No.6

Let $R$ be a ring with identity, $X$ be the set of all nonzero, nonunits of $R$ and $G$ be the group of all units of $R$. A ring $R$ is called $unit$-$duo$ $ring$ if $[x]_{\ell} = [x]_{r}$ for all $x \in X$ where $[x]_{\ell} = \{ux \,|\, u \in G\}$ (resp. $[x]_{r} = \{xu \,|\, u \in G\}$) which are equivalence classes on $X$. It is shown that for a semisimple unit-duo ring $R$ (for example, a strongly regular ring), there exist a finite number of equivalence classes on $X$ if and only if $R$ is artinian. By considering the zero divisor graph (denoted $\widetilde{\Gamma} (R)$) determined by equivalence classes of zero divisors of a unit-duo ring $R$, it is shown that for a unit-duo ring $R$ such that $\widetilde{\Gamma} (R)$ is a finite graph, $R$ is local if and only if diam($\widetilde{\Gamma}(R)$) = 2.

Rings With The Symmetric Property for Idempotent-Products

한준철,심효섭 영남수학회 2018 East Asian mathematical journal Vol.34 No.5

Let R be a ring with the unity 1, and let e be an idempo- tent of R. In this paper, we discuss some symmetric property for the set {(a1, a2, ..., an) ∈ Rn : a1a2 · · · an = e}. We here investigate some properties of those rings with such a symmetric property for an arbitrary idempotent e; some of our results turn out to generalize some known re- sults observed already when n = 2 and e = 0, 1 by several authors. We also focus especially on the case when n = 3 and e = 1. As consequences of our observation, we also give some equivalent conditions to the commutativity for some classes of rings, in terms of the symmetric property.

RINGS WITH A FINITE NUMBER OF ORBITS UNDER THE REGULAR ACTION

한준철,박상원 대한수학회 2014 대한수학회지 Vol.51 No.4

Let R be a ring with identity, X(R) the set of all nonzero, non-units of R and G(R) the group of all units of R. We show that for a matrix ring Mn(D), n ≥ 2, if a,b are singular matrices of the same rank, then |oℓ(a)| = |oℓ(b)|, where oℓ(a) and oℓ(b) are the orbits of a and b, respectively, under the left regular action. We also show that for a semisimple Artinian ring R such that X(R) ≠ ∅, R [수식], with Di infinite division rings of the same cardinalities or R is isomorphic to the ring of 2×2 matrices over a finite field if and only if |oℓ(x)| = |oℓ(y)| for all x,y ∈ X(R).

The general linear group over a ring

한준철 대한수학회 2006 대한수학회보 Vol.43 No.3

Letm be any positive integer, R be a ring with iden-tity, M m (R) be the matrix ring of all m by m matrices over Rand Gm (R) be the multiplicative group of all m by m nonsingularmatrices in M m (R). In this paper, the following are investigated:(1) for any pairwise coprime ideals fI1;I2;:;I n g in a ring R,M m (R=(I1\ I2\\ In )) is isomorphic toM m (R=I 1) M m (R=I 2)M m (R=I n ), and so Gm (R=(I1 \ I2 \ \ In )) is isomorphicto Gm (R=I 1) Gm (R=I 2) Gm (R=I n ); (2) In particular, ifR is a nite ring with identity, then the order of Gm (R) can becomputed.