线性代数课程思政教学案例的设计与实践

在线性代数课程中进行课程思政教学案例的设计,应以课程内容为源头,综合运用学习的课堂、文化的课堂、实践的课堂.在学习的课堂,通过深入挖掘知识点,帮助学生理解线性代数知识体系中蕴含的马克思主义哲学观;在文化的课堂,通过数学史料的讲解,帮助学生树立文化自信,激发爱国情怀;在实践的课堂,通过应用实例的分析和实践,激励学生学以致用,报效祖国.     

关于UR-环

A ring is said to be UR if every element can be written as the sum of a unit and a regular element. These rings are shown to be a unifying generalization of regular rings, clean rings and （S, 2）-ring~. Some relations of these rings are studied and several properties of clean rings and （S, 2）-rings are extended. PAng extensions of UR-rings are also investigated.     

关于环上矩阵的广义逆

本文得到了一般带有对合反自同构的结合环上一类矩阵{1,…,i}-逆存在的充要条件,给出了{3}-逆,{4}-逆,{1,…,i}-逆的表式,而这类矩阵则概括了左右主理想整环,单Artin环(特别是体)上所有矩阵.     

Integrating multiple images in a sampled phase-only hologram

Past research has demonstrated that a sampled phase-only hologram(SPOH) is capable of representing an image without the magnitude component of the hologram. At present, an SPOH can only record and reconstruct a single source image. In this Letter, we propose, for the first time, to the best of our knowledge, a method for representing multiple images with a single integrated SPOH(ISPOH). Subsequently, each image can be retrieved from the ISPOH with a unique key parameter and displayed as a visible image on a phase-only spatial light modulator.     

复矩阵和的Drazin逆

1引 言设Cn×n为n×n复矩阵的集合,对A∈Cn×n,满足rank(Ak+1 )=rank( Ak)的最小非负整数k称为A的指标,记为Ind(A)=κ,则存在唯一矩阵AD∈Cn×n,满足下列矩阵方程组[1]:Ak=Ak+1AD AD=ADAAD AAD=ADAAD称为A的Drazin逆.若Ind(A)=1,则AD称为A的群逆,记为A#.显然Ind(A)=0当且仅当A非奇异,此时AD =A-1.     

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．     

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.     

正则环和GP-内射环

A ring R is called left Gp-injective if for any a∈R, there exists a positive integer n such that any left R-homomorphism of Raⁿ into R extends to one of R into R. In this paper, we prove that the centre of semiprime (left nonsingular) left GP- injective ring is regular ring, and improve some propositions in [3].     

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。