Euclidean modules, submodules of their direct sums and strong reduced echelon form

Abstract

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.

36pages