环Z/p~kZ上矩阵广义逆的拓展

设R=Z/pkZ(其中k>1,p是一个奇素数),A是R上一个给定的可相似对角化的n阶矩阵.利用组合方法和有限局部环上的矩阵方法,讨论了矩阵A的拓展广义逆,得到了矩阵A的拓展广义逆存在的充要条件和一些的计数定理.     

Quasipolar Property of Generalized Matrix Rings

The article concerns the question of when a generalized matrix ring K _s (R) over a local ring R is quasipolar. For a commutative local ring R, it is proved that K _s (R) is quasipolar if and only if it is strongly clean. For a general local ring R, some partial answers to the question are obtained. There exist noncommutative local rings R such that K _s (R) is strongly clean, but not quasipolar. Necessary and sufficient conditions for a single matrix of K _s (R) (where R is a commutative local ring) to be quasipolar is obtained. The known results on this subject in [5 Cui , J. , Chen , J. ( 2011 ). When is a 2 × 2 matrix ring over a commutative local ring quasipolar? Comm. Alg. 39 : 3212 – 3221 .[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]] are improved or extended.     

On the Generalized Strongly Nil-Clean Property of Matrix Rings

Let R be an associative unital ring and not necessarily commutative.We analyze conditions under which every n × n matrix A over R is expressible as a sum A =E1 +…+ Es + N of (commuting) idempotent matrices Ei and a nilpotent matrix N.     

交换环上Rao正则矩阵的广义Cramer法则

Gong Z,Aldeen M和Elsner L在[A note on a generalized Cramer’s rule,Linear AlgebraApp.,2002,340:253-254]中给出结论:对任意的k,α∈Qk,n,β∈Qk,m有|Xα,β|=|A-1|AYαβ,其中A∈n×n可逆矩阵,AX=Y.本文给出交换环上Rao正则矩阵的广义Cramer法则.     

Triangular Matrix Representations of Rings of Generalized Power Series

Let R be a ring and S a cancellative and torsion-free monoid and 〈 a strict order on S. If either （S,≤） satisfies the condition that 0 ≤ s for all s ∈ S, or R is reduced, then the ring [[R^S,≤]] of the generalized power series with coefficients in R and exponents in S has the same triangulating dimension as R. Furthermore, if R is a PWP ring, then so is [[R^S,≤]].     

Generalized Inverses of Matrices over Rings

Let R be a ring, * be an involutory function of the set of all finite matrices over R. In this paper, necessary and sufficient conditions are given for a matrix to have a (1,3)-inverse, (1,4)-inverse, or Moore-P enrose inverse, relative to *. Some results about generalized inverses of matrices over division rings are generalized and improved.     

环上矩阵广义Moore-Penrose逆的存在性

讨论带有对合反自同构*有单位元的结合环R上矩阵的广义Moore-Penrose 逆,给出了环R上矩阵的广义Moore-Penrose逆存在的几个充要条件．特别,得到了环 R上矩阵A的关于M和N的广义Moore-Penrose逆存在的充要条件是A有分解A= GDH,其中D2=D,(MD)*=MD,(GD)*MGD+M(I-D)和DHN-1(DH)*+ (I-D)M-1均可逆．     

广义幂级数环上的PS模

Let R be a commutative ring and(S,≤)a strictly totally ordered monoid which satisfies the condition that 0≤s for every s ∈ S,In this paper we show that if RM is a PS-module,then the module [[M^s,≤]]of generalized power series over M is a PS [[R^s,≤]]-module.     

Generalized PP and Zip Subrings of Matrix Rings

Let R be an abelian ring. We consider a special subring An, relative to α2,…, αn∈ REnd（R）, of the matrix ring Mn（R） over a ring R. It is shown that the ring An is a generalized right PP-ring （right zip ring） if and only if the ring R is a generalized right PP-ring （right zip ring）. Our results yield more examples of generalized right PP-rings and right ziu rings.     

Coherence and Generalized Morphic Property of Triangular Matrix Rings

Let R be a ring. R is left coherent if each of its finitely generated left ideals is finitely presented. R is called left generalized morphic if for every element a in R, l(a) = Rb for some b ∈ R, where l(a) denotes the left annihilator of a in R. The main aim of this article is to investigate the coherence and the generalized morphic property of the upper triangular matrix ring T _n (R) (n ≥ 1). It is shown that R is left coherent if and only if T _n (R) is left coherent for each n ≥ 1 if and only if T _n (R) is left coherent for some n ≥ 1. And an equivalent condition is obtained for T _n (R) to be left generalized morphic. Moreover, it is proved that R is left coherent and left Bézout if and only if T _n (R) is left generalized morphic for each n ≥ 1.     

正则环上矩阵的加权广义逆

定义了正则环上矩阵的加权广义逆,得到其存在的充要条件和表达式,推广了以往文献的相应结果.     

本原分式 Artin 的 Exchange 环上矩阵环

文章证明了：如果Ｒ或要为本原Ａｒｔｉｎ的Ｅｘｃｈａｎｇｅ环,则Ｍｎ（Ｒ）≌（Ｓ）当且仅当Ｒ≌Ｓ。     

On Skew Triangular Matrix Rings

Let α be a nonzero endomorphism of a ring R, n be a positive integer and T_n(R, α) be the skew triangular matrix ring. We show that some properties related to nilpotent elements of R are inherited by T_n(R, α). Meanwhile, we determine the strongly prime radical, generalized prime radical and Behrens radical of the ring R[x; α]/(x~n), where R[x; α] is the skew polynomial ring.     

有强法式的体上矩阵

设D为除环,A∈Dn×n,则可用初等变换将λI-A化简为对角阵A= diag(1,…,1,φ1,…,φr),其中(?)i为D上首1多项式并且φ1|…|φr.如果这个对角阵A在形状上是唯一的,则称A是有强法式的矩阵.本文应用中心原子因子与初等因子给出了体上有强法式的矩阵的本质刻画,给出了体上矩阵有强法式的一些充要条件.     

形式三角矩阵环的Gorenstein遗传性和FP-内射性

我们研究了形式三角矩阵环上模的Gorenstein(半遗传)遗传性,有限表现性和FP-内射性.给出了形式三角矩阵环是Gorenstein(半遗传)遗传的充要条件,并得出了形式三角矩阵环是n-FC环的充分条件.     

主理想整环上n阶矩阵环中的Goldbach问题

Abstract: In this paper, we consider the Goldbach's problem for matrix rings, namely, we decompose an n ×n (n > 1) matrix over a principal ideal domain R into a sum of two matrices in Mn(R) with given determinants. We prove the following result: Let n > 1 be a natural number and A = (αij) be a matrix in Mn(R). Define d(A) := g.c.d{αij}. Suppose that p and q are two elements in R. Then (1) If n > 1 is even, then A can be written as a sum of two matrices X, Y in Mn(R) with det(X) = p and det(Y) = q if and only if d(A) |p-q; (2) If n > 1 is odd, then A can be written as a sum of two matrices X, Y in Mn(R) with det(X) = p and det(Y) = q if and only if d(A) |p + q. We apply the result to the matrices in Mn(Z) and Mn(Q[x]) and prove that if R = Z or Q[x], then any nonzero matrix A in Mn(R) can be written as a sum of two matrices in Mn(R) with prime determinants.     

矩阵环中的基于幺半群的极大的Armendariz子环

Let M be a monoid. Maximal M-Armendariz subrings of upper triangular matrix rings are identified when R is M-Armendariz and reduced. Consequently, new families of M- Armendariz rings are presented.     

斜Armendariz矩阵环

Let R be a ring.We show in the paper that the subring Un(R) of the upper triangular matrix ring Tn(R) is α-skew Armendariz if and only if R is α-rigid,also it is maximal in some non α-skew Armendariz rings,where α is a ring endomorphism of R with α(1) = 1.     

广义四元数体上矩阵的最小多项式

本文给出了广义四元数体上方阵的最小多项式与最小中心多项式的构造公式，讨论了它们的性质及其应用，得到广义四元数方阵相似于对角矩阵的一个充要条件。