• Communications of the Korean Mathematical Society
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• v.23 no.4
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• pp.469-477
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• 2008

We introduce the notion of two-sided $\Gamma$-$\alpha$-derivation of a $\Gamma$-near-ring and give some generalizations of [1, 2].

• The Pure and Applied Mathematics
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• v.21 no.4
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• pp.237-246
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• 2014

Let M be a prime ${\Gamma}$-ring and let d be a derivation of M. If there exists a fixed integer n such that $(d(x){\alpha})^nd(x)=0$ for all $x{\in}M$ and ${\alpha}{\in}{\Gamma}$, then we prove that d(x) = 0 for all $x{\in}M$. This result can be extended to semiprime ${\Gamma}$-rings.

• Communications of the Korean Mathematical Society
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• v.24 no.2
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• pp.161-169
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• 2009

In this paper, we introduce the generalized ideal-based zero-divisor graph structure of near-ring N, denoted by $\widehat{{\Gamma}_I(N)}$. It is shown that if I is a completely reflexive ideal of N, then every two vertices in $\widehat{{\Gamma}_I(N)}$ are connected by a path of length at most 3, and if $\widehat{{\Gamma}_I(N)}$ contains a cycle, then the core K of $\widehat{{\Gamma}_I(N)}$ is a union of triangles and rectangles. We have shown that if $\widehat{{\Gamma}_I(N)}$ is a bipartite graph for a completely semiprime ideal I of N, then N has two prime ideals whose intersection is I.