研究了形式三角矩阵半环Tri(R,M,S)的反导子和高阶反导子.证明了形式三角矩阵半环Tri(R,M,S)的每个反导子可由半环R,S的反导子和(R,S)-双半模M中的可反元来刻画;半环Tri(R,M,S)的任一高阶反导子均可由半环R,S的高阶反导子和(R,S)-双半模M中的可反元族来刻画.

研究了形式三角矩阵半环Tri(R,M,S)的Jordan 双导子,给岀了形式三角矩阵半环 Tri(R,M,S)的Jordan双导子的等价刻画,进而证明了在某些条件下形式三角矩阵半环 Tri(R,M,S)的每一个Jordan双导子都是双导子.

探讨了交换整环上矩阵空间保持相似关系和对称矩阵空间保持合同关系的函数,证明了如下结论:设f是交换整环R到自身的一个映射,且当f(1)≠0时f(1)可逆,n(n≥3)是一个整数.则有:1)f是R上n阶矩阵空间的保持相似关系的函数当且仅当f≡c,其中c∈R或f=cδ,其中c∈R且c≠0,δ为R的非零自同态;2)f是R上n阶对称矩阵空间的保持合同关系的函数当且仅当f≡c,其中c∈R或f=cδ,其中c∈R,c≠0,δ为R的非零自同态.

探讨了交换整环上反对称矩阵空间中保持行列式的函数,证明了如下结论:设f是交换整环R到自身的一个映射,n(n≥3)是一个整数.如果n是奇数,那么f是R上n阶反对称矩阵空间的保持行列式的函数当且仅当f是R上的奇函数;如果n是偶数,那么f是R上n阶反对称矩阵空间的保持行列式的函数当且仅当f是R上n阶全矩阵空间的保持行列式的函数当且仅当f=f(1)δ,其中fn(1)=f(1),δ是R上的非零自同态.

设R是一个交换环,f是R到自身的一个映射.如果f保持R上全矩阵空间(或上三角矩阵空间)中的伴随矩阵,则f称为R上全矩阵空间(或上三角矩阵空间)保持伴随矩阵的函数.探讨了交换环上全矩阵空间和上三角矩阵空间保持伴随矩阵的函数,证明了对于交换环R到自身的任一个映射f,下列条件等价:(1)f是R上n阶矩阵空间保持伴随矩阵的函数,(2)厂是R上n阶上三角矩阵空间保持伴随矩阵的函数,(3)f=f(1)δ,其中fn-1(1)=f(1)且δ是R的非零自同态.所得的结果拓广了域上的重要结论.

研究交换半环上加法可消的广义矩阵代数的Jordan导子、导子和反导子,给岀了广义矩阵代数的Jordan导子、导子和反导子的刻画,进而证明了在某些条件下广义矩阵代数的每一个Jordan导子都可表示为一个导子和一个反导子之和.

研究了形式三角矩阵半环Tri(R,M,S)的自同构和反自同构.证明了:半环Tri(R,M,S)的任一自同构可由半环R,S的自同构和(R,S)-双半模M的一个半线性自同构来表示;半环Tri(R,M,S)的任一反自同构可由半环R到S的一个反同构、S到R的一个反同构和(R,S)-双半模M的一个半线性反自同构来表示.

研究形式三角矩阵半环Tri(R,M,S)双导子,给出加法可消交换半环上形式三角矩阵半环双导子的基本性质;讨论这类形式三角矩阵半环Tri(R,M,S)的双导子与半环R,S的双导子及(R,S)-双半模同态之间的关系,从而获得三角矩阵半环双导子的等价刻画.

研究了形式三角矩阵半环Tri(R,M,S)的导子和高阶导子.证明了半环Tri(R,M,S)的任一导子可由半环R,S的导子和(R,S)-双半模M的一个拟同态来表示;半环Tri(R,M,S)的任一高阶导子可由半环R,S的高阶导子和(R,S)-双半模M中满足一定条件的一族可加映射来表示.

探讨了交换环上上三角矩阵空间、对称矩阵空间以及全矩阵空间中保持行列式的函数.证明了如下结论:1)设f是交换环R到自身的一个映射,n(n≥2)是一个整数,则下列条件等价:①f是R上n阶上三角矩阵空间中保持行列式的函数;②f=f(1)δ,其中f(0)=0,f(1)n=f(1),δ满足δ(xy)=δ(x)δ(y).2)设f是交换环R到自身的一个映射,n(n≥3)是一个整数,则下列条件等价:①f是R上n阶对称矩阵空间中保持行列式的函数;②f是R上n阶全矩阵空间中保持行列式的函数;③f=f(1)δ,其中f(1)n=f(1),δ是R上的非零自同态.

研究了欧氏环中元素的最大公因子与最小公倍,利用矩阵的初等变换,给出了欧氏环中多个元素的最大公因子与最小公倍的统一求法.

设R是一个环,f是R到自身的一个映射,Mn(R)和Tn(R)分别是R上n阶矩阵空间和n阶上三角矩阵空间.如果?A∈Mn(R)(或?A∈Tn(R)),映射A→f(A)=(f(aij))保持逆矩阵,则称f为R上n阶矩阵空间(或R上n阶上三角矩阵空间)保持逆矩阵的函数.证明了对于整环R到自身的一个映射f,下列条件等价:1)f是R上n阶矩阵空间保持逆矩阵的函数;2)f是R上n阶上三角矩阵空间保持逆矩阵的函数;3)f=f(1)δ,其中f(1)2=1且δ是R的一个非零自同态.

A square matrix is said to be diagonalizable if it is similar to a diagonal matrix. In this paper, we discuss diagonability of matrices over commutative semirings and give an equivalent condition for an idempotent matrix over a commutative semiring to be diagonalizable. Also, we obtain an equivalent description for a matrix over a multiplicatively cancellative and commutative semiring to be diagonalizable.

In this paper, Dedekind's theorem on the linear independence of homomorphisms of commutative semirings is studied. Furthermore, this theorem is extended to the case of linear independence of compositions of homomorphisms and powers of a derivation. The results obtained in this paper generalize and develop previous results for fields.

Fillmore's Theorem says that if A is a nonscalar matrix of order n over a field F and gamma 1,gamma 2, horizontal ellipsis ,gamma n is an element of F are such that gamma 1+gamma 2+MIDLINE HORIZONTAL ELLIPSIS+gamma n=trA, then there is a matrix B of order n over F similar to A with diagonal (gamma 1,gamma 2, horizontal ellipsis ,gamma n). Borobia extended Fillmore's Theorem to the matrices over the ring Z of integers. In this paper, we will extend Fillmore's Theorem to the matrices over a general factorial ring.

A square matrix over a semiring is called strongly invertible if all of its leading principal submatrices are invertible. In this paper, the strongly invertible matrices over a semiring are discussed and an equivalent condition for a square matrix over a semiring to be strongly invertible is given. Also, some equivalent descriptions are obtained for a semiring over which the product of any two strongly invertible matrices with the same size is strongly invertible. Some of the results obtained in this paper generalize and develop previous results for matrices over the field of complex numbers and matrices over commutative rings.

In this paper, we investigate the free sets and the free subsemimodules in a semimodule over a commutative semiring S. First, we discuss some properties of the free sets and give a sufficient condition for a nonempty finite set to be free in a finitely generated free S-semimodule and obtain a relation between free set and linear independent set in an S-semimodule. Then we consider the free subsemimodules and prove that the rank of any free subsemimodule of a finitely generated S-semimodule M does not exceed that of M. Also, we give some equivalent descriptions for a commutative semiring S to have the property that all nonzero subsemimodules of any finitely generated free S-semimodule are free. Partial results obtained in the paper develop and generalize the corresponding results for modules over rings and linear spaces over fields.

研究了半环上矩阵的Moore-Penrose逆,给出了半环上矩阵Moore-Penrose逆存在的条件及等价刻画;并且在Moore-Penrose逆存在的情况下给出了矩阵Moore-Penrose逆的表达式.

研究了半环上矩阵的双加权广义Moore-Penrose逆.获得了半环上矩阵双加权广义Moore-Penrose逆存在的充要条件及其结构形式.同时讨论了半环上矩阵双加权广义Moore-Penrose逆的唯一性问题.

In this paper, the inner products on the semimodules over a commutative semiring are investigated. Some characterizations for orthogonal sets and standard orthogonal basis in the semimodules are given. In particular, an equivalent description is obtained for a semifield S satisfying the property that every standard orthogonal set in a finitely generated semimoduleMover S can be extended to a standard orthogonal basis forM. Also, the adjoint homomorphisms of the semimodules are discussed and some properties of the adjoint homomorphisms are obtained. Partial results obtained in this paper generalize and develop corresponding results for semilinear spaces of n-dimensional vectors over commutative zerosumfree semirings and for unitary spaces.