Abstract
In this paper the notion of derivations on $\Gamma$-generalized Boolean semiring are established, namely $\Gamma$-$(f, g)$ derivation and $\Gamma$-$(f, g)$ generalized derivation. We also investigate the commutativity of prime $\Gamma$-generalized Boolean semiring admitting $\Gamma$-$(f, g)$ derivation and $\Gamma$-$(f, g)$ generalized derivation satisfying some conditions.
Highlights
We investigate the commutativity of prime Γ-generalized Boolean semiring admitting Γ-( f, g) derivation and Γ-( f, g) generalized derivation satisfying some conditions
There has been a great deal of work concerning commutativity of prime rings and prime near rings with derivations or generalized derivations satisfying certain differential identity (Ali, 2012; Asci, 2007; Bell, 2012; Rehman, 2011; Quadri, 2003)
The concept of Γ-derivations in Γ-near ring was introduced by Jun, Kim and Cho (Jun, 2003)
Summary
There has been a great deal of work concerning commutativity of prime rings and prime near rings with derivations or generalized derivations satisfying certain differential identity (Ali, 2012; Asci, 2007; Bell, 2012; Rehman, 2011; Quadri, 2003). Mason (Bell & Mason, 1987) introduced derivations on Γ-near rings and studied some basic properties. Kazaz and Alkan (Kazaz & Alkan, 2008) introduced the notion of two-side Γ-α derivation of Γ-near rings and investigated some commutativity of prime and semiprime Γ-near rings. In 2011, the notion of derivations in prime Γ-semiring was introduced by M.A. Javed et al ( Javed et al, 2013). Later in 2014, M.R. Khan and M.M. Hasnain (Khan & Hasnain, 2014) introduced the notion of generalized Γ-derivation in Γ-near rings and investigated some basic properties. We introduce the notion of Γ-( f, g) derivations and Γ-( f, g) generalized derivations on Γ-generalized Boolean semirings, and investigate some related properties. We investigate some commutativity results for Γ-generalized Boolean semiring involving Γ-( f, g) derivation and Γ-( f, g) generalized derivation
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