"\\begin{pmatrix}\n 1 & 3 \\\\\n 3 & 2\n\\end{pmatrix}\\in R" and "\\begin{pmatrix}\n 1 & 2 \\\\\n 2 & 1\n\\end{pmatrix}\\in R", but "\\begin{pmatrix}\n 1 & 3 \\\\\n 3 & 2\n\\end{pmatrix}\\begin{pmatrix}\n 1 & 2 \\\\\n 2 & 1\n\\end{pmatrix}=\\begin{pmatrix}\n 7 & 5 \\\\\n 7 & 8\n\\end{pmatrix}\\not\\in R". So "R" is not a subring of "M_2(\\mathbb Z)".
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