• Kyungpook Mathematical Journal
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• v.60 no.3
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• pp.445-453
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• 2020

Let R be a commutative semiring with identity and M be a unitary R-semimodule. Let 𝜙 : 𝒮(M) → 𝒮(M) ∪ {∅} be a function, where 𝒮(M) is the set of all subsemimodules of M. A proper subsemimodule N of M is called 𝜙-prime subsemimodule, if r ∈ R and x ∈ M with rx ∈ N \𝜙(N) implies that r ∈ (N :_R M) or x ∈ N. So if we take 𝜙(N) = ∅ (resp., 𝜙(N) = {0}), a 𝜙-prime subsemimodule is prime (resp., weakly prime). In this article we study the properties of several generalizations of prime subsemimodules.

• Communications of the Korean Mathematical Society
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• v.39 no.1
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• pp.259-266
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• 2024

The purpose of this note is a wide generalization of the topological results of various classes of ideals of rings, semirings, and modules, endowed with Zariski topologies, to r-strongly irreducible r-ideals (endowed with Zariski topologies) of monoids, called terminal spaces. We show that terminal spaces are T₀, quasi-compact, and every nonempty irreducible closed subset has a unique generic point. We characterize rarithmetic monoids in terms of terminal spaces. Finally, we provide necessary and sufficient conditions for the subspaces of r-maximal r-ideals and r-prime r-ideals to be dense in the corresponding terminal spaces.

• 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 $1{\leq}h{\leq}min\{m,n\}$.