• Kyungpook Mathematical Journal
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• 제50권2호
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• pp.329-336
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• 2010

In this paper, we introduce a partitioning subsemimodule of a semimodule over a semiring which is useful to develop the quotient structure of semimodule. Indeed we prove : 1) The quotient semimodule M=N(Q) is essentially independent of choice of Q. 2) If f : M ${\rightarrow}$ M' is a maximal R-semimodule homomorphism, then $M/kerf_{(Q)}\;\cong\;M'$. 3) Every partitioning subsemimodule is subtractive. 4) Let N be a Q-subsemimodule of an R-semimodule M. Then A is a subtractive subsemimodule of M with $N{\subseteq}A$ if and only if $A/N_{(Q{\cap}A)}\;=\;\{q+N:q{\in}Q{\cap}A\}$ is a subtractive subsemimodule of $M/N_{(Q)}$.

• Kyungpook Mathematical Journal
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• 제47권4호
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• pp.473-480
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• 2007

In this paper we define prime right ideals of semirings and prove that if every right ideal of a semiring R is prime then R is weakly regular. We also prove that if the set of right ideals of R is totally ordered then every right ideal of R is prime if and only if R is right weakly regular. Moreover in this paper we also define prime subsemimodule (generalizing the concept of prime right ideals) of an R-semimodule. We prove that if a subsemimodule K of an R-semimodule M is prime then $A_K(M)$ is also a prime ideal of R.

• Kyungpook Mathematical Journal
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• 제45권4호
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• pp.595-602
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• 2005

Here we define Central separable semialgebras and to prove some structure theorems for central separable cancellative, semialgebras over a commutative and cancellative semiring.

• 충청수학회지
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• 제26권1호
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• pp.37-44
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• 2013

Let R be a commutative semiring with $1_R{\neq}0_R$. Characterization of subsemimodules, prime subsemimodules and primary subsemimodules which are subtractive extensions of Q-subsemimodules in R-semimodules are investigated.

• Korean Journal of Mathematics
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• 제27권2호
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• pp.535-545
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• 2019

The main idea of this paper is to introduce the notion of $cat^1$-monoids and to prove that the category of crossed semimodules ${\mathcal{C}}=(A,B,{\partial})$ where A is a group is equivalent to the category of $cat^1$-monoids. This is a generalization of the well known equivalence between category of $cat^1$-groups and that of crossed modules over groups.

• Kyungpook Mathematical Journal
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• 제60권3호
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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.