• 대한수학회논문집
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• 제19권2호
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• pp.233-242
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• 2004

The aim of this note is to study properties of the generalized centroid of the semi-prime gamma rings. Main results are the following theorems: (1) Let M be a semi-prime $\Gamma$-ring and Q a quotient $\Gamma$-ring of M. If W is a non-zero submodule of the right (left) M-module Q, then $W\Gamma$W $\neq 0. Furthermore Q is a semi-prime $\Gamma$-ring. (2) Let M be a semi-prime $\Gamma$-ring and $C_{{Gamma}$ the generalized centroid of M. Then $C_{\Gamma}$ is a regular $\Gamma$-ring. (3) Let M be a semi-prime $\Gamma$-ring and $C_{\gamma}$ the extended centroid of M. If $C_{\gamma}$ is a $\Gamma$-field, then the $\Gamma$-ring M is a prime $\Gamma$-ring.

• Journal of applied mathematics & informatics
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• 제30권3_4호
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• pp.603-612
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• 2012

In this paper, we introduce and study the intuitionistic fuzzy prime (semi-prime) ${\Gamma}$-ideals of ${\Gamma}$-LA-semigroups and some interesting properties are investigated. The main result of the paper is: if $A={\langle}{\mu}_A,{\gamma}_A{\rangle}$ is an IFS in ${\Gamma}$-LA-semigroup S, then $A={\langle}{\mu}_A,{\gamma}_A{\rangle}$ is an intuitionistic fuzzy prime (semi-prime) ${\Gamma}$-ideal of S if and only if for any $s,t{\in}[0,1]$, the sets $U({\mu}_A,s)=\{x{\in}S:{\mu}_A(x){\geq}s\}$ and $L({\gamma}_A,t)=\{x{\in}S:{\gamma}_A(x){\leq}t\}$ are prime (semi-prime) ${\Gamma}$-ideals of S.

• 대한수학회논문집
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• 제24권3호
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• pp.341-351
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• 2009

The notion of rough sets was introduced by Z. Pawlak in the year 1982. The notion of a $\Gamma$-semigroup was introduced by M. K. Sen in the year 1981. In 2003, Y. B. Jun studied the roughness of sub$\Gamma$-semigroups, ideals and bi-ideals in i-semigroups. In this paper, we study rough prime ideals and rough fuzzy prime ideals in $\Gamma$-semigroups.

• 열처리공학회지
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• 제19권5호
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• pp.249-256
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• 2006

Structural studies have been performed on precipitation hardening found in $L1_2$ ordered ${\gamma}^{\prime}-Ni_3(Al,Ti)$ alloys using transmission electron microscopy. A uniform solid solution of ${\gamma}^{\prime}-L1_2$ ordered phase supersaturated with Ni can be obtained by solution annealing in a suitable temperature range. The ${\gamma}^{\prime}$ phase hardens appreciably by the fine precipitation of disordered ${\gamma}$. The shape of ${\gamma}$ precipitates formed during aging is initially spherical or round-cubic and grow into platelets as aging proceeds. High resolution electron microscopy revealed that the ${\gamma}$ precipitates are perfectly coherent with the matrix ${\gamma}^{\prime}$ as long as the ${\gamma}$-precipitates are plates. The loss of coherency initiates by the introduction of dislocations at the ${\gamma}/{\gamma}^{\prime}$ interface followed by the step formation at the dislocations. The ${\gamma}$ precipitates become globular after the loss of coherency. The strength of ${\gamma}^{\prime}-Ni_3(Al,Ti)$ increases over the temperature range of experiment by the precipitation of fine ${\gamma}$ particles. The peak temperature where a maximum strength was obtained shifted to higher temperature.

• Kyungpook Mathematical Journal
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• 제54권4호
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• pp.629-638
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• 2014

The concept of prime and weakly prime ideal in semigroups has been introduced by G. Szasz [4]. In this paper, we define the involution in po-${\Gamma}$-semigroups, then we extend some results on prime, semiprime and weakly prime ideals to the involution po-${\Gamma}$-semigroup S. Also, we characterize intra-regular involution po-${\Gamma}$-semigroups. We establish that in the involution po-${\Gamma}$-semigroup S such that the involution preserves the order, an ideal of S is prime if and only if it is both weakly prime and semiprime and if S is commutative, then the prime and weakly prime ideals of S coincide. Finally, we prove that if S is a po-${\Gamma}$-semigroup with order preserving involution, then the ideals of S are prime if and only if S is intra-regular.

• 대한수학회보
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• 제43권1호
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• pp.9-16
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• 2006

The purpose of this paper is to define the symmetric $bi-(\sigma,\;\tau)$ derivations on prime and semi prime Gamma rings and to prove some results concerning symmetric $bi-(\sigma,\;\tau)$derivations on prime and semi prime Gamma rings.

• 대한수학회보
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• 제44권2호
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• pp.203-213
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• 2007

The aim of this paper is to introduce modules over the generalized of a semi-prime ${\Gamma}$-ring

• Korean Journal of Mathematics
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• 제5권2호
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• pp.135-139
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• 1997

We introduce the concepts of weakly prime and weakly semi-prime ideals in po-${\Gamma}$-semigroup and give some characterizations of weakly prime and weakly semi-prime ideals of po-${\Gamma}$-semigroups analogous to the characterizations of weakly prime and weakly semi-prime ideals of po-semigroups considered by N. Kehayopulu.

• 분석과학
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• 제6권5호
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• pp.515-520
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• 1993

Rene 80 주조블레이드가 고온에서 사용중인 사용 온도에 의해 ${\gamma}^{\prime}$이라는 석출물이 기지(${\gamma}^{\prime}$) 내에서 생성되기 때문에 고온강도는 그 석출물에 좌우된다. 즉, 석출물의 격자상수는 기지의 격자상수보다 크기 때문에 석출물이 생성되거나 성장할 때에 ${\gamma}-{\gamma}^{\prime}$ 격자어긋남(lattice misfit)에 의해 전위가 발생한다고 일반적으로 알려져 있다. 이러한 격자어긋남은 석출물(${\gamma}^{\prime}$)의 크기가 클수록 원형${\rightarrow}$사각형${\rightarrow}$판상형으로 바뀌어 가면서 증가한다. 따라서 본 연구에서는 이러한 격자어긋남의 양을 주사투과전자현미경에서 수렴성 빔전자 회절(CBED)을 사용하여 측정하여, 석출물과 인접한 기지의 전위밀도의 증가가 기지 내에서 석출물이 생성하고 성장함에 따른 격자어긋남이 원인인가를 관찰하였다.

• 대한수학회논문집
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• 제13권2호
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• pp.251-256
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• 1998

We introduce the concept of weakly prime ideals in po-$\Gamma$-semigroup and give some characterizations of weakly prime ideals.