Malaysian Journal of Mathematical Sciences, January 2021, Vol. 15, No. 1
Contributions to Generalized Derivation on Prime Near-Ring with its Application in the Prime Graph
Khan, M. A.
Corresponding Email: moharram.alikhan@umyu.edu.ng
Received date: 22 December 2019
Accepted date: 20 September 2020
Abstract:
In this paper, we discuss the notion of prime near- ring, which was introduced by Bell and Mason (1987) and Wang (1994) independently. Recently, many authors have investigated commutativity of prime and semi-prime rings admitting suitably constrained derivations (see Daif and Bell (1992), Gölbasi and Koç (2009) for references). Daif and Bell (1992) showed that a prime ring R must be commutative if it admits a derivation \(d\) such that either \(d([x,y])=[x,y]\) or \(d([x,y])=-[x,y]\: \forall \: x,y \in I\) where \(I\) is a non-zero ideal of R. Beck (1988) linked a commutative ring R to a graph by using the elements R as vertices and any two vertices \(x\), \(y\) are adjacent if and only if \(x \ne y\) and \(xy=0\). The zero-divisor graph of a commutative ring R is a graph with the set of non-zero zero-divisors of R as the vertices and any two vertices \(x\), \(y\) are adjacent if and only
if \(x \ne y\) and \(xy=0\). Some comparable results on near-rings have also been derived Beidar et al. (1996), Boua and Oukhtite (2011), Gölbasi
and Koç (2009). The prime graph of a near-ring is a graph with vertices as the set of elements of \(N\) and edges as the set of vertex pair \({x,y}\) such that \(xNy=0\). Indeed \(N\) is prime if and only if prime graph is a star graph (see Bhavanari et al. (2010)). The objective of the present paper
is to extend some results on prime rings admitting generalized derivation to prime near-rings, and some results on relationship between the prime
graph and the zero-divisor graph of \(N\). In addition, examples are given to demonstrate the primeness in the hypothesis is not superfluous. Finally,
we pose some open problems.
Keywords: Commutativity, generalized derivation, prime graph, prime near-ring, star graph, zero-divisor graph
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