• Communications of the Korean Mathematical Society
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• v.23 no.1
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• pp.19-27
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• 2008

The motivation mainly comes from the conditions of the (ordered) ideals to be prime or semiprime that are of importance and interest in (ordered) semigroups and in (ordered) $\Gamma$-semigroups. In 1981, Sen [8] has introduced the concept of the $\Gamma$-semigroups. We can see that any semigroup can be considered as a $\Gamma$-semigroup. The concept of ordered ideal extensions in ordered $\Gamma$-semigroups was introduced in 2007 by Siripitukdet and Iampan [12]. Our purpose in this paper is to introduce the concepts of the ordered n-prime ideals and the ordered n-semiprime ideals in ordered $\Gamma$-semigroups and to characterize the relationship between the ordered n-prime ideals and the ordered ideal extensions in ordered $\Gamma$-semigroups.

• Kyungpook Mathematical Journal
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• v.48 no.4
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• pp.585-591
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• 2008

In 1981, Sen [4] have introduced the concept of $\Gamma$-semigroups. We have known that $\Gamma$-semigroups are a generalization of semigroups. In this paper, we introduce the concepts of the extensions of s-prime ideals, prime ideals, s-semiprime ideals and semiprime ideals in $\Gamma$-semigroups and characterize the relationship between the extensions of ideals and some congruences in $\Gamma$-semigroups.

• Kyungpook Mathematical Journal
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• v.54 no.4
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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.

• Communications of the Korean Mathematical Society
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• v.13 no.2
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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.

• Korean Journal of Mathematics
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• v.5 no.2
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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.

• Korean Journal of Mathematics
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• v.17 no.1
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• pp.77-81
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• 2009

We introduce the notions of a left(right) ${\Gamma}$-filter in a po-${\Gamma}$-semigroups and give a characterization of a left(right) ${\Gamma}$-filter of a po-${\Gamma}$-semigroups in term of right(left) prime ${\Gamma}$-ideals.

• Journal of the Korean Mathematical Society
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• v.55 no.6
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• pp.1381-1388
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• 2018

Let ${\Delta}$ be a simplicial complex, $I_{\Delta}$ its Stanley-Reisner ideal and $K[{\Delta}]$ its Stanley-Reisner ring over a field K. Assume that ${\Gamma}(R)$ denotes the zero-divisor graph of a commutative ring R. Here, first we present a condition on two reduced Noetherian rings R and R', equivalent to ${\Gamma}(R){\cong}{\Gamma}(R{^{\prime}})$. In particular, we show that ${\Gamma}(K[{\Delta}]){\cong}{\Gamma}(K^{\prime}[{\Delta}^{\prime}])$ if and only if ${\mid}Ass(I_{\Delta}){\mid}={\mid}Ass(I_{{{\Delta}^{\prime}}}){\mid}$ and either ${\mid}K{\mid}$, ${\mid}K^{\prime}{\mid}{\leq}{\aleph}_0$ or ${\mid}K{\mid}={\mid}K^{\prime}{\mid}$. This shows that ${\Gamma}(K[{\Delta}])$ contains little information about $K[{\Delta}]$. Then, we define the squarefree zero-divisor graph of $K[{\Delta}]$, denoted by ${\Gamma}_{sf}(K[{\Delta}])$, and prove that ${\Gamma}_{sf}(K[{\Delta}){\cong}{\Gamma}_{sf}(K[{\Delta}^{\prime}])$ if and only if $K[{\Delta}]{\cong}K[{\Delta}^{\prime}]$. Moreover, we show how to find dim $K[{\Delta}]$ and ${\mid}Ass(K[{\Delta}]){\mid}$ from ${\Gamma}_{sf}(K[{\Delta}])$.

• Kyungpook Mathematical Journal
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• v.55 no.4
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• pp.827-835
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• 2015

Let U be a ${\sigma}$-square closed Lie ideal of a 2-torsion free ${\sigma}$-prime ${\Gamma}$-ring M. Let $d{\neq}1$ be an automorphism of M such that $[u,d(u)]_{\alpha}{\in}Z(M)$ on U, $d{\sigma}={\sigma}d$ on U, and there exists $u_0$ in $Sa_{\sigma}(M)$ with $M{\Gamma}u_0{\subseteq}U$. Then, $U{\subseteq}Z(M)$. By applying this result, we generalize the results of Oukhtite and Salhi respect to ${\Gamma}$-rings. Finally, for a non-zero derivation of a 2-torsion free ${\sigma}$-prime $\Gamma$-ring, we obtain suitable conditions under which the $\Gamma$-ring must be commutative.

• Journal of applied mathematics & informatics
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• v.30 no.3_4
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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.

• Bulletin of the Korean Mathematical Society
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• v.46 no.6
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• pp.1051-1060
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• 2009

In this paper, we give topological properties of collection of prime ideals in 2-primal near-rings. We show that Spec(N), the spectrum of prime ideals, is a compact space, and Max(N), the maximal ideals of N, forms a compact $T_1$-subspace. We also study the zero-divisor graph $\Gamma_I$(R) with respect to the completely semiprime ideal I of N. We show that ${\Gamma}_{\mathbb{P}}$ (R), where $\mathbb{P}$ is a prime radical of N, is a connected graph with diameter less than or equal to 3. We characterize all cycles in the graph ${\Gamma}_{\mathbb{P}}$ (R).