On Stable Rank of Rings, Ge-Rings and Quasi-Euclidean Rings

Let R be a ring with unity, GL (R) n the group of all invertible n by n matrices over R andGE (R) n the subgroup of GL (R) n generated by invertible elementary n by n matrices over R. If F is a field,then, in linear algebra, GL (F) GE (F) n n for every positive integer n. If R is a Euclidean ring, then it iswell-known that GL (R) GE (R) n n for every positive integer n. In 1966 (Publ. Math. IHES 30 (1966),5-53), P. M. Cohn introduced the concept of a generalized Euclidean ring, i.e., a ring R with unity is called ageneralized Euclidean ring, or GE-ring for short, if and only if GL (R) GE (R) n n for every positive integern. Recently in 2015, 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 by B. Bougaut (1977) and A. Leutbecher (1978), if and only if it is a commutative ring with unityand every pair (b, a) of elements in R has a terminating division chain of finite length starting from it, the pair(b, a) with this property is also called a Euclidean pair by A. Alahmadi, S. K. Jain, T. Y. Lam, and A. Leroy(J. Algebra (2014)). As an example, let E be the ring of all algebraic integers in the field of complex numbers.Then E is a quasi-Euclidean ring, but it is not Euclidean.)The notion of the stable rank of a ring R, denoted by sr(R), was introduced by H. Bass in 1964. Theresults on the stable rank of rings have close relation to the concepts of GE-rings and n GE -rings. Forexample, if sr(R) = 1, then R is a GE-ring. As examples, sr(R) = 1 for every local ring R and every Artinianring R.In this project we will study the relations between the stable rank of rings and the concepts of GEringsand n GE -rings. We also want to identify more examples of rings which are quasi-Euclidean rings butnot Euclidean rings, to identify more examples of rings which are GE-rings but not quasi-Euclidean rings, toidentify more examples of rings which are n GE -rings for some integer n but not GE-rings. (Recall that aring R with unity is called a n GE -ring if GL (R) GE (R) n n for positive integer n.)