Linear Operators That Strongly Preserve Nilpotent Matrices over Antinegative Semirings

Let S be an antinegative commutative semiring without zero divisors and Mn（S） be the semiring of all n × n matrices over S. For a linear operator L on Mn（S）, we say that L strongly preserves nilpotent matrices in Mn（S） if for any A ∈ Mn（S）, A is nilpotent if and only if L（A） is nilpotent. In this paper, the linear operators that strongly preserve nilpotent matrices over S are characterized.

The Semigroup Structure of Left Clifford Semirings

In this paper, we generalize Clifford semirings to left Clifford semirings bymeans of the so-called band semirings. We also discuss a special case of this kind of semirings,that is, strong distributive lattices of left rings.

Semirings which are unions of rings

Sernirings which are a disjoint union of rings form a variety S which contains the variety of all rings and the variety of all idempotent sernirings, and in particular, the variety of distributive lattices. Various structure theorems are established which bring insight into the structure of the lattice of subvarieties of S.

Zero divisors and prime elements of bounded semirings

A semiring is an algebraic structure similar to a ring, but without the requirement that each element must have an additive inverse. A bounded semiring is a semiring equipped with a compatible bounded partial order. In this paper, properties of zero divisors and prime elements of a bounded semiring are studied. In particular, it is proved that under some mild assumption, the set Z（A） of nonzero zero divisors of A is A ／ {0, 1}, and each prime element of A is a maximal element. For a bounded semiring A with Z（A） = A ／ {0, 1}, it is proved that A has finitely many maximal elements if ACC holds either for elements of A or for principal annihilating ideals of A. As an application of prime elements, we show that the structure of a bounded semiring A is completely determined by the structure of integral bounded semirings if either ｜Z（A）｜ = 1 or ｜Z（A）｜ -- 2 and Z（A）2 ≠ 0. Applications to the ideal structure of commutative rings are also considered. In particular, when R has a finite number of ideals, it is shown that the chain complex of the poset I（R） is pure and shellable, where I（R） consists of all ideals of R.

A Hua-like Theorem in Semirings＊

In this paper, we establish a Hua-Like theorem in some kind of semirings. This can be regarded as a generalized version of Hua＇s theorem from rings to semirings.

On Ordered Ideals in Ordered Semirings

An ordered semiring is a semiring S equipped with a partial order ≤ such that the operations are monotonic and constant 0 is the least element of S.In this paper,several notions,for example,ordered ideal,minimal ideal,and maximal ideal of an ordered semiring,simple ordered semirings,etc.,are introduced.Some properties of them are given and characterizations for minimal ideals are established.Also,the matrix semiring over an ordered semiring is consid-ered.Partial results obtained in this paper are analogous to the corresponding ones on ordered semigroups,and on the matrix semiring over a semiring.

Several Classes of Additively Non-Regular Semirings

In this paper, we introduce Green＇s .-relations on semirings and define [left, right] adequate semirings to explore additively non-regular semirings. We characterize the semirings which are strong b-lattices of [left, right] skew-halfrings. Also, as further generalization, the semirings are described which are subdirect products of an additively commutative idempotent semiring and a [left, right] skew-halfring. We extend results of constructions of generalized Clifford semirings （given by M. K. Sen, S. K. MaRy, K. P. Shum, 2005） and the semirings which are subdirect products of a distributive lattice and a ring （given by S. Ghosh, 1999） to additively non-regular semirings.

Complement of the reduced non-zero component graph of free semimodules

Let M be a finitely generated free semimodule over a semiring S with identity having invariant basis number property with a basisα={α1,...,αk}.The complement■of the reduced non-zero component graph■of M,is the simple undirected graph with■as the vertex set and such that there is an edge between two distinct vertices■and■if and only if there exists no i such that both ai,biare non-zero.In this paper,we show that the graph■is connected and find its domination number,clique number and chromatic number.In the case of finite semirings,we determine the degree of each vertex,order,size,vertex connectivity and girth of■.Also,we give a necessary and sufficient condition for■to be Eulerian or Hamiltonian or planar.

浅析服装行业营销模式——以VANCL和SEMIR为例

VANCL作为后起之秀，打败老大PPG，本文尝试以它为例，探讨网络直销成功的营销模式：SEMIR作为中国民营仓业的代表，本文尝试探讨服装行业连锁经营的成功秘诀，并提出对两者的展望。

Rectangular Ring Congruences on an E-inversive Semiring

In this paper, we discussed the property of rectangular band semiring congruence and ring congruence on a semiring and gave some characterizations and structure of rectangular ring congruence on an E-inversive semiring.

On Homomorphism of Valuation Algebras

In this paper, firstly, a necessary condition and a sufficient condition for an isomorphism between two semiring-induced valuation algebras to exist are presented respectively. Then a general valuation homomorphism based on different domains is defined, and the corresponding homomorphism theorem of valuation algebra is proved.

Dual Zariski Spaces of Modules

Let R be a commutative ring with identity,M be an R-module,L(M)denote the set of all submodules of M and g■(M)\{OM}.For any submodule N of M,we set gV^(d)(N)={K∈g:K■N}and gζ^(d)(M)=(GV^(d)(N):N■L(M)).Consider χ■L(R)\{R},where L(R)is the set of all ideals of R.We set χV(I)=(J∈x:I■J)and χζ(R)={xV(I):I■L(R)}for any ideal I of R.In this paper,we investigate when,for arbitrary χ and g as above,χζ(R)and gζ^(d)(M)form a topology and a semimodule,respectively.We investigate the structure of gζ^(d)(M)in the case that it is a semimodule.

The lattice of idempotent distributive semiring varieties

A solution is given for the word problem for free idempotent distributive semirings. Using this solution the lattice L(ID) of subvarieties of the variety ID of idempotent distributive semirings is determined. It turns out that L(ID) is isomorphic to the direct product of a four-element lattice and a lattice which is itself a subdirect product of four copies of the lattice L (B) of all band varieties. Therefore L(ID) is countably infinite and distributive. Every subvariety of ID is finitely based.

Divisibility Proper ties of the Semiring of Ideals of an Integral Domain

Let D be an integral domain,F^(+)(D)(resp.,f^(+)(D))be the set of nonzero(resp.,nonzero finitely generated)ideals of D,R_(1)=f+(D)∪{(0)},and R_(2)=F+(D)∪{(0)}.Then(R_(i),㊉,■)for i=1,2 is a commutative semiring with identity under I㊉J=I+J and I■J=ZJ for all I,J∈R_(i).In this paper,among other things,we show that D is a Priifer domain if and only if every ideal of R_(1)is a k-ideal if and only if R_(1)is Gaussian.We also show that D is a Dedekind domain if and only if R_(2)is a unique factorization semidomain if and only if R_(2)is a principal ideal semidomain.These results are proved in a more general setting of star operations on D.