• Kyungpook Mathematical Journal
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• v.45 no.1
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• pp.21-31
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• 2005

We introduce the notions of nilpotent element, quasi-regular element in a semiring which is a distributive lattice of rings. The concept of Jacobson radical is introduced for this kind of semirings.

• Communications of the Korean Mathematical Society
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• v.21 no.2
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• pp.219-226
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• 2006

We consider matrices whose entries can be viewed as elements of both nonnegative integers and nonnegative rational numbers. We determine the differences of factor rank of matrices over the two semirings.

• 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.

• Korean Journal of Mathematics
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• v.30 no.2
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• pp.341-349
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• 2022

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

• Bulletin of the Korean Mathematical Society
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• v.32 no.2
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• pp.337-342
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• 1995

A semiring is a binary system $(S, +, \times)$ such that (S, +) is an Abelian monoid (identity 0), (S,x) is a monoid (identity 1), $\times$ distributes over +, 0 $\times s s \times 0 = 0$ for all s in S, and $1 \neq 0$. Usually S denotes the system and $\times$ is denoted by juxtaposition. If $(S,\times)$ is Abelian, then S is commutative. Thus all rings are semirings. Some examples of semirings which occur in combinatorics are Boolean algebra of subsets of a finite set (with addition being union and multiplication being intersection) and the nonnegative integers (with usual arithmetic). The concepts of matrix theory are defined over a semiring as over a field. Recently a number of authors have studied various problems of semiring matrix theory. In particular, Minc [4] has written an encyclopedic work on nonnegative matrices.

• Kyungpook Mathematical Journal
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• v.47 no.4
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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.

• The Pure and Applied Mathematics
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• v.12 no.4 s.30
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• pp.253-274
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• 2005

An additive regular Semiring S is left inversive if the Set E+ (S) of all additive idempotents is left regular. The set LC(S) of all left inversive semiring congruences on an additive regular semiring S is a lattice. The relations $\theta$ and k (resp.), induced by tr. and ker (resp.), are congruences on LC(S) and each $\theta$-class p$\theta$ (resp. each k-class pk) is a complete modular sublattice with $p_{min}$ and $p_{max}$ (resp. With $p^{min}$ and $p^{max}$), as the least and greatest elements. $p_{min},\;p_{max},\;p^{min}$ and $p^{max}$, in particular ${\epsilon}_{max}$, the maximum additive idempotent separating congruence has been characterized explicitly. A semiring is quasi-inversive if and only if it is a subdirect product of a left inversive and a right inversive semiring.

• Journal of the Korean Mathematical Society
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• v.45 no.4
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• pp.1043-1056
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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 ordered pairs which satisfy multiplicative properties with respect to spanning column rank of matrices over semirings.

• Journal of the Korean Mathematical Society
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• v.32 no.4
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• pp.849-856
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• 1995

There are some papers on structure theorems for the spaces of matrices over certain semirings. Beasley, Gregory and Pullman [1] obtained characterizations of semiring rank 1 matrices over certain semirings of the nonnegative reals. Beasley and Pullman [2] also obtained the structure theorems of Boolean rank 1 spaces. Since the semiring rank of a matrix differs from the column rank of it in general, we consider a structure theorem for semiring rank in [1] in view of column rank.

• 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 $l$.