• Communications of the Korean Mathematical Society
• /
• v.19 no.2
• /
• pp.233-242
• /
• 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
• /
• v.30 no.3_4
• /
• pp.603-612
• /
• 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.

• Communications of the Korean Mathematical Society
• /
• v.24 no.3
• /
• pp.341-351
• /
• 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.

• Journal of the Korean Society for Heat Treatment
• /
• v.19 no.5
• /
• pp.249-256
• /
• 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
• /
• v.54 no.4
• /
• pp.629-638
• /
• 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.

• Bulletin of the Korean Mathematical Society
• /
• v.43 no.1
• /
• pp.9-16
• /
• 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.

• Bulletin of the Korean Mathematical Society
• /
• v.44 no.2
• /
• pp.203-213
• /
• 2007

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

• Korean Journal of Mathematics
• /
• v.5 no.2
• /
• pp.135-139
• /
• 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.

• Analytical Science and Technology
• /
• v.6 no.5
• /
• pp.515-520
• /
• 1993

By the real use of Rene 80 cast blades at high temperature ${\gamma}^{\prime}$ precipitates in the matrix(${\gamma}$) mainly due to the operating temperature. These precipitates play main role for strenthening of the blades. Generally known that dislocation density increases due to ${\gamma}-{\gamma}^{\prime}$ mismatch by the generation and growth of the precipitates, because the lattice parameter of ${\gamma}^{\prime}$ is higher than that of ${\gamma}$. These lattice parameters of ${\gamma}$ and ${\gamma}^{\prime}$ are determined through the CBED(Convergent Beam Electron Diffraction) method by STEM(Scanning Transmission Electron Microscope) in this work. And also studied, whether and how much the dislocation density increases by the generation and growth of the precipitates.

• Communications of the Korean Mathematical Society
• /
• v.13 no.2
• /
• pp.251-256
• /
• 1998

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