• Korean Journal of Mathematics
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• v.27 no.3
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• pp.779-791
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• 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
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• v.59 no.1
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• pp.35-45
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• 2019

The ${\Gamma}$-semihyperring is a generalization of the concepts of a semiring, a semihyperring and a ${\Gamma}$-semiring. Here, the notions of (strongly) regular ${\Gamma}$-semihyperring, idempotent ${\Gamma}$-semihyperring; invertible set, invertible element in a ${\Gamma}$-semihyperring are introduced, and several examples given. It is proved that if all subsets of ${\Gamma}$-semihyperring are strongly regular then for every ${\Delta}{\subseteq}{\Gamma}$, there is a ${\Delta}$-idempotent subset of R. Regularity conditions of ${\Gamma}$-semihyperrings in terms of ideals of ${\Gamma}$-semihyperrings are also characterized.

• Journal of the Chungcheong Mathematical Society
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• v.35 no.2
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• pp.115-124
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• 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 Korean Mathematical Society
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• v.51 no.3
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• pp.593-607
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• 2014

Let R be an I-semiring and S(R) be the set of all identity-summand elements of R. In this paper we introduce the total graph of R with respect to identity-summand elements, denoted by T(${\Gamma}(R)$), and investigate basic properties of S(R) which help us to gain interesting results about T(${\Gamma}(R)$) and its subgraphs.

• Journal of the Korean Mathematical Society
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• v.51 no.1
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• pp.189-202
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• 2014

An element r of a commutative semiring R with identity is said to be identity-summand if there exists $1{\neq}a{\in}R$ such that r+a = 1. In this paper, we introduce and investigate the identity-summand graph of R, denoted by ${\Gamma}(R)$. It is the (undirected) graph whose vertices are the non-identity identity-summands of R with two distinct vertices joint by an edge when the sum of the vertices is 1. The basic properties and possible structures of the graph ${\Gamma}(R)$ are studied.