欧几里得模、其直和的子模及强简化阶梯形
Euclidean modules, submodules of their direct sums and strong reduced echelon form
浏览论文内容
中文总结 AI 辅助
本文研究环上的欧几里得模及其直和子模，引入强简化阶梯形，并证明欧几里得模类广泛存在于多种经典代数构造中。
中文摘要 AI 辅助
本文旨在研究环上的欧几里得模和l-有限欧几里得模、其有限直和的子模，以及系数在欧几里得模和双模中的矩阵的简化阶梯形和强简化阶梯形。欧几里得模的概念是欧几里得环概念的模论类比。欧几里得环（不一定交换）的类是一个性质良好但“非常小”的环类，具有类似算术的性质。相反，令人惊讶的是，欧几里得模的类是一个“大类”的模。对于定义在任意环上的许多环的构造，欧几里得模都存在。例如，对于每个系数在任意环$D$中的斜多项式环$A=D[x;\s, \d]$、广义外尔代数$D[x,y;\s, a]$、外尔代数$A_1=K[x][\der; \frac{d}{dx}]$、$U(\sl2)$以及许多经典代数，都存在欧几里得模。
英文摘要
The aim of the paper is to study Euclidean and l-finite Euclidean modules over a ring, submodules of their finite direct sums, reduced and strong reduced echelon forms of matrices with coefficients in Euclidean modules and bimodules. The concept of Euclidean module is a module-theoretic analogue of the concept of Euclidean ring. The class of (not necessarily commutative) Euclidean rings is a nice but `very small' class of rings that have arithmetic-like properties. On the contrary, surprisingly, the class of Euclidean modules is a `large class' of modules. The Euclidean modules exist for many {\em constructions} of rings that are defined over {\em arbitrary} rings. For example, for every skew polynomial ring $A=D[x;\s, \d]$ with coefficients in an {\em arbitrary} ring $D$, the generalized Weyl algebra $D[x,y;\s, a]$, the Weyl algebra $A_1=K[x][\der; \frac{d}{dx}]$, $U(\sl2)$ and many classical algebras, there exist Euclidean modules.
