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 A1=K[x][der; d/dx], U(sl2) and many classical algebras, there exist Euclidean modules.
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BibTeX
@misc{bavula2026euclidean,
title = {{Euclidean modules, submodules of their direct sums and strong reduced echelon form}},
author = {V. V. Bavula},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.37893},
eprint = {2609.37893},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.37893},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Euclidean modules, submodules of their direct sums and strong reduced echelon form
AU - V. V. Bavula
PY - 2026
DA - 2026/09/29
PB - arXiv
DO - 10.48550/arxiv.2609.37893
UR - https://arxiv.org/abs/2609.37893
AB - 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; d/dx], U(sl2) and many classical algebras, there exist Euclidean modules.
ER -