# On Generalized Euclidean Rings and Quasi-Euclidean Rings

## Project Details

### Description

In linear algebra every invertible n by n matrix over a field F can be expressed as a finite product ofelementary matrices over F. This property does not hold in general for any invertible n by n matrixover an arbitrary ring with unity. Let R be a ring with unity and denote by GL (R) n the group of allinvertible n by n matrices over R. Denote by GE (R) n the subgroup of GL (R) n generated byinvertible elementary n by n matrices over R. It is well-known that GL (R) n = GE (R) n for everypositive integer n if R is a Euclidean ring. In 1966 (Publ. Math. IHES 30 (1966), 5-53), P. M. Cohnintroduced the concept of a generalized Euclidean ring, i.e., a ring R with unity is called a generalizedEuclidean ring, or GE-ring for short, if and only if GL (R) n = GE (R) n for every positive integer n.Recently, we prove that a ring R is a GE-ring if it is a quasi-Euclidean ring which is an anothergeneralization of the concept of a Euclidean ring. (Recall that a ring R is a quasi-Euclidean ring,introduced in 1976, if and only if it is a commutative ring with unity and every pair (b, a) of elementsin R has a terminating division chain of finite length starting from it, in other words, a greatest commondivisor of the pair (b, a) exists in R and it can be obtained by applying a terminating division chain offinite length starting from (b, a).)We have known some examples of quasi-Euclidean rings which are not Euclidean rings and alsoknown a few examples of GE-rings which are not quasi-Euclidean rings. To my knowledge, thetheories of generalized Euclidean rings and quasi-Euclidean rings, as part of the theory of Euclideanrings, are still underdeveloped. In this project we want to search for more examples which arequasi-Euclidean rings but not Euclidean rings, and for more examples which are GE-rings but notquasi-Euclidean rings.
Status Finished 1/08/16 → 31/07/17

### UN Sustainable Development Goals

In 2015, UN member states agreed to 17 global Sustainable Development Goals (SDGs) to end poverty, protect the planet and ensure prosperity for all. This project contributes towards the following SDG(s):

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