线性代数中几个疑难问题的处理

基于东南大学线性代数课程的教学实践,本文针对递归数列与行列式、矩阵的初等列变换与列向量组等价、条件的独立性以及线性代数与解析几何的联系这四个方面的问题,提出了处理方案.     

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.     

关于左A—内射环

本文主要证明了：（1）适合右零化子升链条件的左A－内射环为QF环。（2）适合左零化子升链条件的左f-内射环为QF环。（3）若对环R的任意左理想A，B和右理想I满足r(A∩B）=r(A) r(B),rι(I)=I，则R为半完全环且有本质左基座，特别地，右CF的左A－内射环（或E(RR)为投射左R－模）为QF环。     

I-semi-π-regular Rings

Let R be a ring and I an ideal of R. A ring R is called I-semi-π--regular if R/I is π-regular and idempotents of R can be strongly lifted modulo I. Characterizations of I-semi-π-regular rings are given and relations between semi-π-regular rings and semiregular rings are explored.     

相对复盖与包络

本文旨在给出相对复盖与包络的特征刻划．作为应用,证明了：若Ｒ为任意环,０ＡＢＣ０为左Ｒ 模正合列并且Ａ与Ｃ都有平坦复盖,则Ｂ有平坦复盖．     

关于环上矩阵的群逆与Drazin逆

本文给出了环上一类方阵有群逆,｛１,５｝－道的充要条件及其它们的表式,推广了体（域）上关于群逆的Ｃｌｉｎｅ定理．此外还首次得到了矩阵有Ｄｒａｚｉｎ逆的判别准则及其它的表式．     

环上矩阵的群逆与Drazin逆

本文给出了环上一类方阵有群逆与 Drazin 逆的充要条件及其它们的显式,且此类方阵概括了左右主理想整环上所有方阵。     

关于(m,n)-凝聚模与预包络

In this paper,let m,n be two fixed positive integers and M be a right R-module,we define (m,n)-M-flat modules and (m,n)-coherent modules.A right R-module F is called (m,n) M-flat if every homomorphism from an (n,m)-presented right R-module into F factors through a module in addM.A left S-module M is called an (m,n)-coherent module if MR is finitely presented,and for any (n,m)-presented right R-module K,Horn(K,M) is a finitely generated left S-module,where S = End(MR).We mainly characterize (m,n)-coherent modules in terms of preenvelopes (which are monomorphism or epimorphism) of modules.Some properties of (m,n)-coherent rings and coherent rings are obtained as corollaries.     

相对半投射和直投射模

In this paper, the ＂relative＂ version of semi-projectivity is considered. Let M and N be modules. N is called M-semi-projective, if any homomorphism from N to an M-eyclie submodule f（M） of N can be factored through to a homomorphism from N to M and f, where f∈[M, N]. Some properties of relative semi-projectivity are obtained. Next, we consider a wider class of ＂elatively semi-projective＂ modules such as ＂elatively direct-projective＂ modules which were introduced by Nicholson and Zhou. Several properties of their homomorphisms are also obtained.     

关于Drazin逆的刻划

Let ■ be a pre-additive category. Assume that ψ:X→X is a morphism of (?). In this paper, we give the necessary and sufficient conditions for ψ to have the Drazin inverse by using the von Neumann regular inverse for the ψk, and extend a result by Puystjens and Hartwig from the group inverse to Drazin inverse.