# Is the class of commutative generalized Euclidean rings stable under quotient and localization?

Let $$R$$ be an associative ring with identity and let $$E_n(R)$$ be the subgroup of $$GL_n(R)$$ generated by matrices obtained from the identity matrix by replacing an off-diagonal entry by some $$r∈R$$. Let $$D_n(R)$$ be the group of invertible diagonal matrices; note that it normalizes $$E_n(R)$$. The ring $$R$$ is termed a $$GE_n$$-ring if $$GL_n(R)=E_n(R)D_n(R)$$. If $$R$$ is commutative, the latter condition is equivalent to $$SL_n(R) = E_n(R)$$ (to see this, apply Whitehead lemma).

We say that $$R$$ is a generalized Euclidean ring in the sense of P. M. Cohn , or concisely a $$GE$$-ring, if it is a $$GE_n$$-ring for all $$n>1$$. The class of $$GE$$-rings is not stable under quotient since free associative algebras over a field are $$GE$$-rings [Theorem 3.4, 1] whereas free commutative algebras over a field (i.e., polynomial rings over a field) on more than one indeterminate aren't [Proposition 7.3, 1].

Question 1: Is there an example of a commutative $$GE$$-ring with a non-$$GE$$ quotient?

The class of Euclidean rings (and more generally the class of commutative quasi-Euclidean rings ) is known to be stable under localization.

Question 2: Is the class of (commutative) $$GE$$-rings stable under localization?

Note that $$R = \mathbb{Z}[X^{\pm 1}]$$ has a localization which is not a $$GE_2$$-ring by [Corollary 3, 2]. So, if $$R$$ turns out to be a $$GE_2$$-ring (see the corresponding MO post), then $$R$$ would be a counter-example for Question 2.

 On the structure of the $$GL_2$$ of a ring, P. M. Cohn, 1966.
 $$SK_1$$ of an interesting principal ideal domain, D. R. Grayson, 1981.
 Euclidean pairs and quasi-Euclidean rings, A. Alahmadi et al., 2014.

• Hope ok to put in comments. For comm. rings, local -> clean -> stable range 1 -> GE. Burgess,Raphael [J. Alg. Appl. 7 (2008) 195-209] have an example of a local reduced ring such that its classical ring of quotients is not clean. Knox, Levy, McGovern, Shapiro [J. Alg. and its Appl., 8 (2009) ] home.fau.edu/wmcgove1/web/Papers/klms.pdf also provide an example. Are there examples of stable range 1 rings with localizations that are not stable range 1? Estes and Ohm [J.Alg. 7, 343-362 (1967)] look at conditions which yield that localization does not change the stable range. Jun 5, 2016 at 17:08