On Generalized Euclidean Rings and Quasi-Euclidean Rings

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.