• 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.

• Kyungpook Mathematical Journal
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• v.52 no.1
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• pp.91-97
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• 2012

Let $R$ be a commutative semiring. We define a proper ideal $I$ of $R$ to be 2-absorbing (resp., weakly 2-absorbing) if $abc{\in}I$ (resp., $0{\neq}abc{\in}I$) implies $ab{\in}I$ or $ac{\in}I$ or $bc{\in}I$. We show that a weakly 2-absorbing ideal $I$ with $I^3{\neq}0$ is 2-absorbing. We give a number of results concerning 2-absorbing and weakly 2-absorbing ideals and examples of weakly 2-absorbing ideals. Finally we de ne the concept of 0 - (1-, 2-, 3-)2-absorbing ideals of $R$ and study the relationship among these classes of ideals of $R$.

• 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.

• Journal of applied mathematics & informatics
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• v.7 no.1
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• pp.289-295
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• 2000

In this note, we define a ${Q^*}-ideal$ in a seminear-ring which is analogous of a Q-ideal in a semiring, and we construct a quotient seminear-ring. Also, We prove the fundamental theorem of homomorphisms for seminear-rings.

• Kyungpook Mathematical Journal
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• v.56 no.1
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• pp.107-120
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• 2016

Let R be a commutative semiring. The purpose of this note is to investigate the concept of 2-absorbing (resp., weakly 2-absorbing) primary ideals generalizing of 2-absorbing (resp., weakly 2-absorbing) ideals of semirings. A proper ideal I of R said to be a 2-absorbing (resp., weakly 2-absorbing) primary ideal if whenever $a,b,c{\in}R$ such that $abc{\in}I$ (resp., $0{\neq}abc{\in}I$), then either $ab{\in}I$ or $bc{\in}\sqrt{I}$ or $ac{\in}\sqrt{I}$. Moreover, when I is a Q-ideal and P is a k-ideal of R/I with $I{\subseteq}P$, it is shown that if P is a 2-absorbing (resp., weakly 2-absorbing) primary ideal of R, then P/I is a 2-absorbing (resp., weakly 2-absorbing) primary ideal of R/I and it is also proved that if I and P/I are weakly 2-absorbing primary ideals, then P is a weakly 2-absorbing primary ideal of R.

• Kyungpook Mathematical Journal
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• v.54 no.4
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• pp.619-627
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• 2014

Let $\mathcal{M}(S)$ denote the set of all $m{\times}n$ matrices over a semiring S. For $A{\in}\mathcal{M}(S)$, zero-term rank of A is the minimal number of lines (rows or columns) needed to cover all zero entries in A. In [5], the authors obtained that a linear operator on $\mathcal{M}(S)$ preserves zero-term rank if and only if it preserves zero-term ranks 0 and 1. In this paper, we obtain new characterizations of linear operators on $\mathcal{M}(S)$ that preserve zero-term rank. Consequently we obtain that a linear operator on $\mathcal{M}(S)$ preserves zero-term rank if and only if it preserves two consecutive zero-term ranks k and k + 1, where $0{\leq}k{\leq}min\{m,n\}-1$ if and only if it strongly preserves zero-term rank h, where $1{\leq}h{\leq}min\{m,n\}$.

• Journal of the Korean Mathematical Society
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• v.50 no.1
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• pp.127-136
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• 2013

The term rank of a matrix A is the least number of lines (rows or columns) needed to include all the nonzero entries in A. In this paper, we obtain a characterization of linear transformations that preserve term ranks of matrices over antinegative semirings. That is, we show that a linear transformation T from a matrix space into another matrix space over antinegative semirings preserves term rank if and only if T preserves any two term ranks $k$ and $l$.

• Journal of the Korean Mathematical Society
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• v.45 no.2
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• pp.301-312
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• 2008

The spanning column rank of an $m{\times}n$ matrix A over a semiring is the minimal number of columns that span all columns of A. We characterize linear operators that preserve the sets of matrix pairs which satisfy additive properties with respect to spanning column rank of matrices over semirings.

• Journal of applied mathematics & informatics
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• v.42 no.2
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• pp.417-432
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• 2024

In this research article, we study a class of semirings with involution. Differential identities involving two or three derivations of a semiring with second kind involution are investigated. It is analyzed that how these identities, with a special role for second kind involution, bring commutativity to semirings.

• Communications of the Korean Mathematical Society
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• v.31 no.4
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• pp.657-665
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• 2016

In this investigation we studied completely quasi primary and weakly completely quasi primary ideals in ternary semirings. Some characterizations of completely quasi primary and weakly completely quasi primary ideals were obtained. Moreover, we investigated relationships between completely quasi primary and weakly completely quasi primary ideals in ternary semirings. Finally, we obtained necessary and sufficient conditions for a weakly completely quasi primary ideal to be a completely quasi primary ideal.