Invertible Matrices over Finite Additively Idempotent Semirings

We investigate invertible matrices over finite additively idempotent semirings. The main result provides a criterion for the invertibility of such matrices. We also give a construction of the inverse matrix and a formula for the number of invertible matrices.     

交换半环上商半模的若干研究

在经典代数的基础上,讨论了交换半环上的同余关系与同余类,并在它们与可消半模,稠密子半模,序关系及其最小元的基础上,证明了L-商半模的一些性质,然后定义了L-商半模上的不变子半模,并开始在其基础上讨论L-商变换,结合同态同构来研究它的若干性质.     

On Flat Semimodules over Semirings

The following analog of the characterization of flat modules has been obtained for the variety of semimodules over a semiring R: A semimodule _RA is flat (i.e., the tensor product functor – A preserves all finite limits) iff A is L-flat (i.e., A is a filtered colimit of finitely generated free semimodules). We also give new (homological) characterizations of Boolean algebras and complete Boolean algebras within the classes of distributive lattices and Boolean algebras, respectively, which solve two problems left open in [14]. It is also shown that, in contrast with the case of modules over rings, in general for semimodules over semirings the notions of flatness and mono-.atness (i.e., the tensor product functor – A preserves monomorphisms) are different.     

交换半环上三角矩阵代数的自同构

证明了如果尺是含有恒等元的交换半环，那么R上的n阶上三角矩阵代数的自同构都是内自同构．     

Proper/Residually-Finite Idempotent Semirings

An idempotent semiring (= ISR) is called L-E if its underlying additive Abelian semigroup is generated by join-primes. Not all ISRs are L-E; not even when finite. The submodule of “linear-recognizable” elements of an ISR $\mathcal {M}$ is denoted $\mathcal {C}\mathcal {M}$ , and $\mathcal {M}$ is called proper if there are enough elements in $\mathcal {C}\mathcal {M}$ to separate points. If there are enough finite-index congruences to separate points, $\mathcal {M}$ is called residually-finite. Finite and proper ISRs are always residually-finite, but finite ISRs are not always proper, unless they are L-E. For certain classes of ISRs, conditions are given to guarantee proper and residual-finiteness. Among these is one which requires that the compact elements of the linear dual of $\mathcal {M}$ belong to $\mathcal {C}\mathcal {M}$ . Another condition requires that the recognizable subsets of a certain underlying monoid remain recognizable under the closure operator relative to a certain natural topology. These conditions are automatic for any finite L-E ISR, or any L-E ISR arising from a bounded, distributive lattice. Thus, a large class of proper/residually-finite ISRs exists. Moreover, the theorem of Malcev for semigroups (finitely-generated, commutative implies residually-finite) is shown to fail for ISRs in general.     

几类加法非正则半环

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.     

On V-semirings and Semirings All of Whose Cyclic Semimodules Are Injective

In this article, we introduce and study V- and CI-semirings—semirings all of whose simple and cyclic, respectively, semimodules are injective. We describe V-semirings for some classes of semirings and establish some fundamental properties of V-semirings. We show that all Jacobson-semisimple V-semirings are V-rings. We also completely describe the bounded distributive lattices, Gelfand, subtractive, semisimple, and antibounded, semirings that are CI-semirings. Applying these results, we give complete characterizations of congruence-simple subtractive and congruence-simple antibounded CI-semirings which solve two earlier open problems for these classes of CI-semirings.     

Completely Prime Ideals in Multiplicatively Idempotent Semirings

Mathematical Notes - Structure properties of multiplicatively idempotent semirings are considered. Basic results concerning completely prime ideals in multiplicatively idempotent semirings are...     

某些非负交换半环上强保持矩阵M-P逆的线性算子(英)

设S是无零因子的非负交换半环或者有限生成布尔代数,μ_n（S）记S上的矩阵集合．本文确定了μ_n（S）上的强保持矩阵M－P逆的线性算子半群N_n（S）的结构．     

Idempotent Units of Commutative Group Rings

We find a necessary and sufficient condition for every normalized unit in a commutative unitary group ring to be an idempotent unit. Our criterion reduces the general situation to the torsion case. This extends in some way results due to Karpilovsky [7 Karpilovsky , G. ( 1982 ). On units in commutative group rings . Arch. Math. (Basel) 38 : 420 – 422 . [Google Scholar], 8 Karpilovsky , G. ( 1983 ). On finite generation of unit groups of commutative group rings . Arch. Math. (Basel) 40 : 503 – 508 . [Google Scholar]] and Danchev [1-3 Danchev , P. V. ( 2008 ). Trivial units in commutative group algebras . Extr. Math. 23 : 49 – 60 . Danchev , P. V. ( 2009 ). Trivial units in abelian group algebras . Extr. Math. 24 : 47 – 53 . Danchev , P. V. ( 2009 ). Idempotent units in commutative group rings . Kochi J. Math. 4 : 61 – 68 . ].     

一类幂等半环的性质和结构

研究了加法半群为半格的半环类S+l中的乘法带半环和矩形带半环类BR中的乘法带半环;给出了ID半环中乘法带半环的结构定理,即ID∩.■°D=.■z∨.■z∨D.     

Adjoints and Monads Related to Compact Lattices and Compact Lawson Idempotent Semimodules

For compact Hausdorff Lawson lattices L that are idempotent semirings with lower semicontinuous multiplications, free compact Hausdorff Lawson L-idempotent semimodules are constructed over compacta and over compact Hausdorff Lawson lower semilattices. It is shown that these free objects are spaces of L-valued regular measures. Related left adjoint functors and monads are described.     

On Polynomial Functions over Finite Commutative Rings

Let R be an arbitrary finite commutative local ring. In this paper, we obtain a necessary and sufficient condition for a function over R to be a polynomial function. Before this paper, necessary and sufficient conditions for a function to be a polynomial function over some special finite commutative local rings were obtained.