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Let $K$ be a principal ideal ring. How to prove that for any $ x= (x_1, x_2)^t \in K^2 $ there exists a matrix $G \in SL_2(K)$ such that $Gx = (\gcd(x_1, x_2),0)^t $ ?

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Do you mean greatest common divisor ? Its not quite clear. –  Rene Schipperus Jun 23 at 12:22
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@ReneSchipperus Yes. –  Shemhamforasch Jun 23 at 12:25
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Remark that the matrix appears in the bottom right corner of the table constructed by the Extended Euclidean algorithm - see here. –  Bill Dubuque Jun 23 at 14:19

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up vote 6 down vote accepted

If it's the greatest common divisor $d$ of $x_{1}$ and $x_{2}$ you mean, and you assume that $x_{1}$ and $x_{2}$ are not both zero, so that $d \ne 0$, then there are $a, b \in K$ such that $a x_{1} + b x_{2} = d$, and now take $$ G = \begin{bmatrix} a & b\\ -x_{2}/d & x_{1}/d \end{bmatrix} $$

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Of course if $x_{1} = x_{2} = 0$, any $G$ will do. –  Andreas Caranti Jun 23 at 12:56

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