Question

(that NOT worked !)

I give then just the review to try in MR:

and the beginning of the review:

if $\text{GL}_n(R)=\text{GE}_n(R)$ for all $n$. As is well-known, classical Euclidean rings are of this type. In particular, $\text{GE}_2$-rings are those for which $\text{GL}_2(R)=\text{GE}_2(R)$. For an arbitrary ring $R$, set $E(a)=\left( \matrix a & 1 \ -1 & 0 \endmatrix \right)$, $a\in R$.

These matrices, together with those in $D_2(R)$, generate the group $\text{GE}2(R)$. The author gives a set of relations connecting these generators, namely, (i) $E(x)E(0)E(y)=-E(x+y)$, $E(\alpha)E(\alpha^{-1})E(\alpha)=-\text{diag}(\alpha,\alpha^{-1})$, $E(x)\cdot\text{diag}(\alpha,\beta)=\text{diag}(\beta,\alpha)\cdot E(\beta^{-1}x\alpha)$, where $x,y\in R$, and $\alpha,\beta\in U(R)$. Other relations may possibly hold as well. Call the ring $R$ universal for GE${\text 2}$ if the above relations, together with those relations valid in $D_2(R)$, are a full set of defining relations for $\text{GE}_2(R)$. For arbitrary $R$, relations (1) already imply that $E_2(R)\Delta\text{GE}_2(R)$, and that every element of $\text{GE}_2(R)$ is expressible in a certain standard form (ii) $\text{diag}(\alpha,\beta)\cdot E(a_1)\cdots E(a_r)$, where $\alpha,\beta\in U(R)$, $a_1,\cdots,a_r\in R$, and some minor restrictions are satisfied. If each element of $\text{GE}2(R)$ has a unique standard form, it follows easily that $R$ is universal for GE${\text 2}$. The converse is false, as is shown by example. Indeed, there is even an intermediate class of rings, namely those for which no non-trivial standard form (ii) can be equal to the identity matrix; the author calls such rings quasi-free for GE${\text 2}$. (2) The property of being a GE-ring is preserved under direct sums, but not under direct products. In order to discuss free products, some auxiliary definitions are required. (a) Let $K$ be a skewfield. Call $R$ a $K$-ring if there is a canonical injection of $K$ into $R$. (b) A semifir (semi-free-ideal-ring) is an integral domain $R$, not necessarily commutative, such that every finitely generated right ideal of $R$ is free, and such that any two bases of a free $R$-module have the same cardinality. These rings were previously studied by the author [J. Algebra 1 (1964), 47--69; MR0161891 (28 #5095)], where they were called "local firs''. (c) A strong GE-ring is a semifir which is a GE-ring. It is shown that $R$ is a strong GE-ring if and only if for each $n\geq 1$, and each relation $\sum{i=1}^na_ib_i=0$, $a_i,b_i\in R$, with $b_1,\cdots,b_n$ not all zero, there exists a matrix $C\in E_n(R)$ such that $(a_1,\cdots,a_n)C$ has at least one zero entry. Rings satisfying this condition had been called "generalized Euclidean'' by H. Bass [ibid. 1 (1964), 367--373; MR0178032 (31 #2290)]. By slightly modifying the arguments in his above mentioned work, the author proves the following theorem: Let $K$ be a skewfield, ${R_\lambda}$ a family of $K$-rings each of which is a strong GE-ring; then the free product of the ${R_\lambda}$ over $K$ is again a strong GE-ring. A special case of this theorem is as follows: The group algebra (over a field) of a free group is a GE-semifir. The same holds for the semigroup algebra of a free semigroup. W. Klingenberg [Arch. Math. 13 (1962), 73--81; MR0143817 (26 #1367)] proved that a local ring $R$ (not necessarily commutative) is a GE-ring. The author shows that $R$ is universal for GE$_{\text 2}$. Further, if $R$ is local but not a skewfield, then there is no unique standard form (2) for the elements of $\text{GE}2(R)$; indeed, in this case $R$ is not even quasi-free for GE${\text 2}$. (3) Call $R$ a discretely normed ring if there is a realvalued norm $|\ |$ on $R$ satisfying $|0|=0$, $|x|\geq 1$ for $x\in R$, $x\neq 0$; $|xy|=|x|\,|y|$, $|x+y|\leq|x|+|y|$, and in addition there is no $x\in R$ for which $1<|x|<2$. Basic lemma: Let $r>1$, and let $a_1,\cdots,a_r$ be nonzero elements of the discretely normed ring $R$ such that none of $a_2,\cdots,a_r$ is a unit; then the norm of the $(1,1)$-entry of $E(a_1)E(a_2)\cdots E(a_r)$ cannot be less than the norm of its $(1,2)$-entry. (Conditions for equality are given, under further hypotheses.) This lemma implies that every discretely normed ring is quasi-free, and hence universal for GE$_{\text 2}$. For such a ring $R$, every involution $(\neq\pm I$) in $\text{GE}2(R)$ is conjugate therein to a matrix $\pm\left( \matrix 1 & 0 \ h & -1 \endmatrix \right)$, with suitably restricted $h\in R$. This implies that a 2-torsion-free discretely normed Dedekind ring which is a GE${\text 2}$-ring must in fact be a principal ideal domain. (a) The ring $R_d$ of all algebraic integers in $Q(\surd-d)$ is discretely normed. Here, $d$ is a square-free positive integer, $d\neq 1,2,3,7,11$. (For these exceptional values, $R_d$ is already known to be Euclidean with respect to the usual norm.) (b) Rings with degree functions, such as $K[X_1,X_2,\cdots,X_n]$ ($K=$ field), are discretely normed. (4) The reviewer and others have asked whether every principal ideal domain must be a GE-ring. The author gives a negative answer. In terms of the notation in (3a) above, it turns out that $R_{19}$ (which is known to be a principal ideal domain) is not a GE$_{\text 2}$-ring. Analogous results are obtained for function fields. For rings with a degree function, it is shown that each element of $\text{GE}2(R)$ has a unique standard form. Not every such ring is a GE${\text 2}$-ring, however. Thus, $K[X_1,\cdots,X_n]$ is a GE-ring if and only if $n=1$. Instead of taking $R$ to be discretely normed, much the same purpose is achieved by assuming $R$ to be a totally ordered ring satisfying some additional restrictions. Such rings are universal for GE$_{\text 2}$. It is shown that $Z[X_1,\cdots,X_n]\ (n>0)$ cannot be a GE${\text 2}$-ring, although it is universal for GE${\text 2}$. (5) For any group $G$, let $G'$ denote its commutator subgroup, and set $G^a=G/G'$. The author proves that $\text{GE}_2(R)/E_2(R)\cong U(R)^a$ if $R$ is universal for GE$_{\text 2}$. Further, $E_2(R)=\text{GE}_2(R)'$ whenever there exist $\alpha,\beta\in U(R)$ with $\alpha+\beta=1$. Specific formulas are obtained for $E_2(R)^a$, under various hypotheses on $R$. For example, if $R$ is a discretely normed ring whose only units are $\pm 1$, then $E_2(R)^a\cong R/M$, where $M$ is the additive subgroup generated by 12. To quote another result of this nature: Let $R$ be any ring which is quasi-free for GE$_{\text 2}$, and let $N$ be the ideal of $R$ generated by ${\alpha-1\colon\alpha\in U(R)}$; then $E_2(R)/\text{GE}_2(R)'\cong R/N$. When $R=Z$, this yields the isomorphism $\text{SL}_2(Z)/\text{GL}_2(Z)'\cong Z/(2)$, a result due to L. K. Hua and the reviewer [Trans. Amer. Math. Soc. 71 (1951), 331--348; MR0043847 (13,328f)]. (6) The author proves that if $U(R)$ is finitely generated (f.g.), and if $R$ is a f.g. $U(R)$-bimodule, then $\text{GE}2(R)$ is also f.g. Conversely, if

$R$ is quasi-free for GE${\text 2}$, and $\text{GE}_2(R)$ is f.g., then $U(R)^a$ is f.g., and $R$ is f.g. as $U(R)$-bimodule. This generalizes the result due to H. Nagao [J. Inst. Polytech. Osaka City Univ. Ser. A 10 (1959), 117--121; MR0114866 (22 #5684)] that if $K$ is a field, and $X$ an indeterminate, then $\text{GL}2(K[X])$ is not f.g. (7) For $m\in Z$, $m\geq 2$, it is known that the matrices $\left( \matrix 1 & m \ 0 & 1 \endmatrix \right)$, $\left( \matrix 1 & 0 \ m & 1 \endmatrix \right)$ generate a free group. Generalizing this, the author proves: Let $R$ be quasi-free for GE${\text 2}$, and let $A,B$ be additive subgroups of $R$ which contain no units; then the group generated by $$ \left{\left( \matrix 1 & a \ 0 & 1 \endmatrix \right)\colon a\in A\right}\cup\left{\left( \matrix 1 & 0 \ b & 1 \endmatrix \right)\colon b\in B\right} $$ is the free product of the group generated by the first set with that generated by the second set. (8) Consider next the question of homomorphisms between linear groups. By a $U$-homomorphism $\varphi$ of the ring $R$ into the ring $S$ is meant an additive homomorphism $x\rightarrow x'$ such that $1'=1$, and $(\alpha a\beta)'=\alpha'a'\beta'$, $a\in R$, $\alpha,\beta\in U(R)$. For a $U$-antihomomorphism $\psi$, replace the last condition by $(\alpha a\beta)'=\beta'a'\alpha'$. The form of the set of relations (i) then implies: If $R$ is universal for GE$_{\text 2}$, then any $U$-homomorphism $\varphi\colon R\rightarrow S$ induces a homomorphism $\varphi^\ast\colon\text{GE}_2(R)\rightarrow\text{GE}_2(S)$, by letting $E(x)\rightarrow E(x')$, $\text{diag}(\alpha,\beta)\rightarrow\text{diag}(\alpha',\beta')$. Likewise, a $U$-antihomomorphism $\psi\colon R\rightarrow S$ induces a homomorphism $\psi^\ast\colon\text{GE}_2(R)\rightarrow\text{GE}_2(S)$ by letting $E(x)\rightarrow E(x')^{-1}$, $\text{diag}(\alpha,\beta)\rightarrow\text{diag}(\alpha',\beta')^{-1}$. These results generalize those of the reviewer for the case where $R=K[X]$, $K$ a field [see Proc. Amer. Math. Soc. 8 (1957), 1111--1113; MR0095860 (20 #2358); Ann. of Math. (2) 66 (1957), 461--466; MR0095882 (20 #2380)]. (9) The author takes up finally the problem of determining all isomorphisms $\text{GL}_n(R)\cong\text{GL}_n(S)$, and as usual, the case $n=2$ presents the greatest difficulty. The results obtained here generalize theorems of Schreier-van der Waerden, Dieudonné, Hua, and the reviewer. A central homothety is an endomorphism of $\text{GL}_n(R)$ given by $A\rightarrow\sigma(A)A$, where $\sigma$ is a homomorphism of $\text{GL}n(R)$ into the group of central units of $R$. The major result for $n=2$ is as follows: Let $R$ be a $K$-ring, $S$ a $K'$-ring, both with a degree function, where $K$ and $K'$ are skewfields of the same characteristic, and where $S$ is a GE${\text 2}$-ring; then every isomorphism $\text{GL}_2(R)\cong\text{GL}_2(S)$ is gotten by taking either $\varphi^\ast$ or $\psi^\ast$ ($\varphi=U$-isomorphism, $\psi=U$-anti-isomorphism), followed by a central homothety and an inner automorphism. The author uses this to handle the case $n>2$, and proves the following. Let $R$ be a $K$-ring, $S$ a $K'$-ring, where $K$ and $K'$ are skewfields of characteristic $\neq 2$. Assume that both $R$ and $S$ have a degree function, and that every f.g. projective $S$-module is free. Then every isomorphism $\text{GL}_n(R)\cong\text{GL}_n(S)$ for $n\geq 3$ is obtained by taking either $\varphi^\ast$ or $\psi^\ast$ (where now dots$\varphi\colon R\rightarrow S$ is a ring isomorphism, and $\psi\colon R\rightarrow S$ a ring anti-isomorphism), followed by a central homothety and an inner automorphism.Reviewed by I. Reiner