• 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.

• Korean Journal of Mathematics
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• v.10 no.2
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• pp.117-122
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• 2002

In this paper, we give some properties of left(right) semi-regular and $g$-regular $po$-semigroups.

• 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.

• Kyungpook Mathematical Journal
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• v.64 no.1
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• pp.171-184
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• 2024

In this paper, we show that if G is strongly regular then the Gallai graph Γ(G) and the anti-Gallai graph ∆(G) of G are edge-regular. We also identify conditions under which the Gallai and anti-Gallai graphs are themselves strongly regular, as well as conditions under which they are 2-connected. We include also a number of concrete examples and a discussion of spectral properties of the Gallai and anti-Gallai graphs.

• 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.

• Journal of applied mathematics & informatics
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• v.24 no.1_2
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• pp.261-271
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• 2007

The symbol C($m_1^{n_1}m_2^{n_2}{\cdots}m_s^{n_s}$) denotes a 2-regular graph consisting of $n_i$ cycles of length $m_i,\;i=1,\;2,\;{\cdots},\;s$. In this paper, we give some construction methods of cyclic($K_v$, G)-designs, and prove that there exists a cyclic($K_v$, G)-design when $G=C((4m_1)^{n_1}(4m_2)^{n_2}{\cdots}(4m_s)^{n_s}\;and\;v{\equiv}1(mod\;2|G|)$.

• Bulletin of the Korean Mathematical Society
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• v.43 no.1
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• pp.125-140
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• 2006

Let $g(n,\;l_1,\;l_2,\;d,\;t,\;q)$ be the number of general4-regular graphs on n labelled vertices with $l_1+2l_2$ loops, d double edges, t triple edges and q quartet edges. We use inclusion and exclusion with five types of properties to determine the asymptotic behavior of $g(n,\;l_1,\;l_2,\;d,\;t,\;q)$ and hence that of g(2n), the total number of general 4-regular graphs where $l_1,\;l_2,\;d,\;t\;and\;q\;=\;o(\sqrt{n})$, respectively. We show that almost all general 4-regular graphs are 2-connected. Moreover, we determine the asymptotic numbers of general 4-regular graphs with given connectivities.

• East Asian mathematical journal
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• v.33 no.3
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• pp.257-263
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• 2017

Let n be any positive integer and ${\mathbb{Z}}_n=\{0,1,{\cdots},n-1\}$ be the ring of integers modulo n. Let $X_n$ be the set of all nonzero, nonunits of ${\mathbb{Z}}_n$, and $G_n$ be the group of all units of ${\mathbb{Z}}_n$. In this paper, by investigating the regular action on $X_n$ by $G_n$, the following are proved : (1) The number of orbits under the regular action (resp. the number of annihilators in $X_n$) is equal to the number of all divisors (${\neq}1$, n) of n; (2) For any positive integer n, ${\sum}_{g{\in}G_n}\;g{\equiv}0$ (mod n); (3) For any orbit o(x) ($x{\in}X_n$) with ${\mid}o(x){\mid}{\geq}2$, ${\sum}_{y{\in}o(x)}\;y{\equiv}0$ (mod n).

• Bulletin of the Korean Mathematical Society
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• v.25 no.2
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• pp.203-209
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• 1988

In this paper, we define a g-regular semigroup which is a generalization of a regular semigroup. And we want to find some properties of g-regular semigroup. G-regular semigroups contains the variety of all regular semigroup and the variety of all periodic semigroup. If a is an element of a semigroup S, the smallest left ideal containing a is Sa.cup.{a}, which we may conveniently write as $S^{I}$a, and which we shall call the principal left ideal generated by a. An equivalence relation l on S is then defined by the rule alb if and only if a and b generate the same principal left ideal, i.e. if and only if $S^{I}$a= $S^{I}$b. Similarly, we can define the relation R. The equivalence relation D is R.L and the principal two sided ideal generated by an element a of S is $S^{1}$a $S^{1}$. We write aqb if $S^{1}$a $S^{1}$= $S^{1}$b $S^{1}$, i.e. if there exist x,y,u,v in $S^{1}$ for which xay=b, ubv=a. It is immediate that D.contnd.q. A semigroup S is called periodic if all its elements are of finite order. A finite semigroup is necessarily periodic semigroup. It is well known that in a periodic semigroup, D=q. An element a of a semigroup S is called regular if there exists x in S such that axa=a. The semigroup S is called regular if all its elements are regular. The following is the property of D-classes of regular semigroup.group.

• Bulletin of the Korean Mathematical Society
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• v.37 no.3
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• pp.487-491
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• 2000

Let G be a covering graph of G. We show that the line graph of G covers the line graph of G. Moreover, if the first covering is regular, then the line-graph covering is regular.