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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 [1], 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 [3]) 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.


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

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    $\begingroup$ 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. $\endgroup$ – W.McGovern Jun 5 '16 at 17:08

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