• Korean Journal of Mathematics
• /
• v.27 no.3
• /
• pp.779-791
• /
• 2019

In this paper, we consider derivation of an ordered ${\Gamma}$-semiring and introduce the notion of reverse derivation on ordered ${\Gamma}$-semiring. Also, we obtain some interesting related properties. Let I be a nonzero ideal of prime ordered ${\Gamma}$-semiring M and let d be a nonzero derivation of M. If ${\Gamma}$-semiring M is negatively ordered, then d is nonzero on I.

• Kyungpook Mathematical Journal
• /
• v.46 no.2
• /
• pp.153-167
• /
• 2006

We study permuting tri-derivations in ${\Gamma}$-rings and give an example.

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

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

• Korean Journal of Mathematics
• /
• v.26 no.2
• /
• pp.175-189
• /
• 2018

In this paper, at first we generalize the notion of algebra over a field. A ${\Gamma}$-algebra is an algebraic structure consisting of a vector space V, a groupoid ${\Gamma}$ together with a map from $V{\times}{\Gamma}{\times}V$ to V. Then, on every associative ${\Gamma}$-algebra V and for every ${\alpha}{{\in}}{\Gamma}$ we construct an ${\alpha}$-Lie algebra. Also, we discuss some properties about ${\Gamma}$-Lie algebras when V and ${\Gamma}$ are the sets of $m{\times}n$ and $n{\times}m$ matrices over a field F respectively. Finally, we define the notions of ${\alpha}$-derivation, ${\alpha}$-representation, ${\alpha}$-nilpotency and prove Engel theorem in this case.

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

• 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.

• Communications of the Korean Mathematical Society
• /
• v.15 no.3
• /
• pp.469-479
• /
• 2000

We define and study the extended centroid of a prime $\Gamma$-ring.

• Journal of the Optical Society of Korea
• /
• v.19 no.3
• /
• pp.237-240
• /
• 2015

We derived the average bit error rate (BER) of coherent free-space optical (FSO) systems with digital binary phase shift keying (BPSK) modulations over atmospheric turbulence channels with a gamma-gamma distribution. To obtain a generalized derivation in a closed-form expression, we used special integrals and transformations of the Meijer G function. Furthermore, we numerically analyzed and simulated the average BER behavior according to the average SNR for different turbulence strengths. Simulation results are demonstrated to confirm the analytical results.

• Journal of the Chungcheong Mathematical Society
• /
• v.35 no.2
• /
• pp.115-124
• /
• 2022

In this paper, we introduce the notion of orthogonal reserve derivation on semiprime 𝚪-semirings. Some characterizations of semiprime 𝚪-semirimgs are obtained by means of orthogonal reverse derivations. We also investigate conditions for two reverse derivations on semiprime 𝚪-semiring to be orthogonal.

• Journal of the Chungcheong Mathematical Society
• /
• v.34 no.2
• /
• pp.119-129
• /
• 2021

In this paper, we introduce the concept of a generalized right f-derivation associated with a derivation d and a function f in 𝚪-incline algebras and give some properties of 𝚪-incline algebras. Also, the concept of d-ideal is introduced in a 𝚪-incline algebra with respect to right f-derivations.