• Journal of applied mathematics & informatics
• /
• 제28권1_2호
• /
• pp.311-324
• /
• 2010

We prove that a regular ordered semigroup S is left simple if and only if every intuitionistic fuzzy left ideal of S is a constant function. We also show that an ordered semigroup S is left (resp. right) regular if and only if for every intuitionistic fuzzy left(resp. right) ideal A = <$\mu_A$, $\gamma_A$> of S we have $\mu_A(a)\;=\;\mu_A(a^2)$, $\gamma_A(a)\;=\;\gamma_A(a^2)$ for every $a\;{\in}\;S$. Further, we characterize some semilattices of ordered semigroups in terms of intuitionistic fuzzy left(resp. right) ideals. In this respect, we prove that an ordered semigroup S is a semilattice of left (resp. right) simple semigroups if and only if for every intuitionistic fuzzy left (resp. right) ideal A = <$\mu_A$, $\gamma_A$> of S we have $\mu_A(a)\;=\;\mu_A(a^2)$, $\gamma_A(a)\;=\;\gamma_A(a^2)$ and $\mu_A(ab)\;=\;\mu_A(ba)$, $\gamma_A(ab)\;=\;\gamma_A(ba)$ for all a, $b\;{\in}\;S$.

• 충청수학회지
• /
• 제28권4호
• /
• pp.561-573
• /
• 2015

The aim of this paper is to introduce the notion of left-right (resp. right-left) symmetric bi-derivation of BCH-algebras and some related properties are investigated.

• 한국수학교육학회지시리즈B:순수및응용수학
• /
• 제12권4호
• /
• pp.253-274
• /
• 2005

An additive regular Semiring S is left inversive if the Set E+ (S) of all additive idempotents is left regular. The set LC(S) of all left inversive semiring congruences on an additive regular semiring S is a lattice. The relations $\theta$ and k (resp.), induced by tr. and ker (resp.), are congruences on LC(S) and each $\theta$-class p$\theta$ (resp. each k-class pk) is a complete modular sublattice with $p_{min}$ and $p_{max}$ (resp. With $p^{min}$ and $p^{max}$), as the least and greatest elements. $p_{min},\;p_{max},\;p^{min}$ and $p^{max}$, in particular ${\epsilon}_{max}$, the maximum additive idempotent separating congruence has been characterized explicitly. A semiring is quasi-inversive if and only if it is a subdirect product of a left inversive and a right inversive semiring.

• East Asian mathematical journal
• /
• 제25권4호
• /
• pp.433-440
• /
• 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.

• Journal of applied mathematics & informatics
• /
• 제27권5_6호
• /
• pp.1447-1457
• /
• 2009

In this paper we define intuitionistic fuzzy interior ideals in ordered semigroups. We prove that in regular(resp. intra-regular and semisimple) ordered semigroups the concepts of intuitionistic fuzzy interior ideals and intuitionistic fuzzy ideals coincide. We prove that an ordered semi group is intuitionistic fuzzy simple if and only if every intutionistic fuzzy interior ideal is a constant function. We characterize intra-regular ordered semi groups in terms of interior (resp. intuitionistic fuzzy interior) ideals.

• Journal of applied mathematics & informatics
• /
• 제42권2호
• /
• pp.333-351
• /
• 2024

In this paper, the concept of Pythagorean fuzzy sets are used to describe in semigroups. Then, some characterizations of regular (resp., intra-regular) semigroups by means of Pythagorean fuzzy left (resp., right) ideals and Pythagorean fuzzy (resp., generalized) bi-ideals of semigroups are investigated. Furthermore, the class of both regular and intra-regular semigroups by the properties of many kinds of their Pythagorean fuzzy ideals also being studied.

• 대한수학회보
• /
• 제43권2호
• /
• pp.299-307
• /
• 2006

The following results ale extended from P-injective rings to AP-injective rings: (1) R is left self-injective regular if and only if R is a right (resp. left) AP-injective ring such that for every finitely generated left R-module M, $_R(M/Z(M))$ is projective, where Z(M) is the left singular submodule of $_{R}M$; (2) if R is a left nonsingular left AP-injective ring such that every maximal left ideal of R is either injective or a two-sided ideal of R, then R is either left self-injective regular or strongly regular. In addition, we answer a question of Roger Yue Chi Ming [13] in the positive. Let R be a ring whose every simple singular left R-module is Y J-injective. If R is a right MI-ring whose every essential right ideal is an essential left ideal, then R is a left and right self-injective regular, left and right V-ring of bounded index.

• 대한수학회지
• /
• 제45권6호
• /
• pp.1647-1659
• /
• 2008

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. First, we investigate some connected conditions of the zero-divisor graph $\Gamma(R)$ of a noncommutative ring R as follows: (1) if $\Gamma(R)$ has no sources and no sinks, then $\Gamma(R)$ is connected and diameter of $\Gamma(R)$, denoted by diam($\Gamma(R)$) (resp. girth of $\Gamma(R)$, denoted by g($\Gamma(R)$)) is equal to or less than 3; (2) if X is a union of finite number of orbits under the left (resp. right) regular action on X by G, then $\Gamma(R)$ is connected and diam($\Gamma(R)$) (resp. g($\Gamma(R)$)) is equal to or less than 3, in addition, if R is local, then there is a vertex of $\Gamma(R)$ which is adjacent to every other vertices in $\Gamma(R)$; (3) if R is unit-regular, then $\Gamma(R)$ is connected and diam($\Gamma(R)$) (resp. g($\Gamma(R)$)) is equal to or less than 3. Next, we investigate the graph automorphisms group of $\Gamma(Mat_2(\mathbb{Z}_p))$ where $Mat_2(\mathbb{Z}_p)$ is the ring of 2 by 2 matrices over the galois field $\mathbb{Z}_p$ (p is any prime).

• Journal of applied mathematics & informatics
• /
• 제28권5_6호
• /
• pp.1217-1225
• /
• 2010

In this paper we define fuzzy interior $\Gamma$-ideals in ordered $\Gamma$-semigroups. We prove that in regular(resp. intra-regular) ordered $\Gamma$-semigroups the concepts of fuzzy interior $\Gamma$-ideals and fuzzy $\Gamma$-ideals coincide. We prove that an ordered $\Gamma$-semigroup is fuzzy simple if and only if every fuzzy interior $\Gamma$-ideal is a constant function. We characterize intra-regular ordered $\Gamma$-semigroups in terms of interior (resp. fuzzy interior) $\Gamma$-ideals.

• East Asian mathematical journal
• /
• 제25권4호
• /
• pp.441-447
• /
• 2009

Jat and Choudhari defined a near-ring R with left bipotent or right bipotent condition in 1979. Also, we can dene a near-ring R as subcommutative if aR = Ra for all a in R. From these above two concepts it is natural to investigate the near-ring R with the properties aR = $Ra^2$ (resp. $a^2R$ = Ra) for each a in R. We will say that such is a near-ring with (1,2)-potent condition (resp. a near-ring with (2,1)-potent condition). Thus, we can extend a general concept of a near-ring R with (m,n)-potent condition, that is, $a^mR\;=\;Ra^n$ for each a in R, where m, n are positive integers. We will derive properties of near-ring with (1,n) and (n,1)-potent conditions where n is a positive integer, any homomorphic image of (m,n)-potent near-ring is also (m,n)-potent, and we will obtain some characterization of regular near-rings with (m,n)-potent conditions.