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Let $R= \left\{ \begin{bmatrix} a & b \\ b & a \\ \end{bmatrix} : a,b \in \mathbb{Z} \right\} $ be the ring of symmetric integer matrices with same diagonal elements.

Is the ring $R$ isomorphic to $\mathbb{Z} \times \mathbb{Z}$ ?

My thoughts:

The two rings are very similar. They have the same amount of inversible elements, $U(R)= \left\{ \begin{bmatrix} 1 & 0 \\ 0 & 1 \\ \end{bmatrix} , \begin{bmatrix} -1 & 0 \\ 0 & -1 \\ \end{bmatrix} , \begin{bmatrix} 0 & 1 \\ 1 & 0 \\ \end{bmatrix}, \begin{bmatrix} 0 & -1 \\ -1 & 0 \\ \end{bmatrix} \right\} $ and $U(\mathbb{Z} \times \mathbb{Z}) = \{(1,1),(-1,-1),(1,-1),(-1,1)\}$.

I have tried $f:R \rightarrow \mathbb{Z} \times \mathbb{Z}, f\left( \begin{bmatrix} a & b \\ b & a \\ \end{bmatrix}\right)=(a,b) $, which fails to be homorphism, due to multiplication.

I have also tried $g:R \rightarrow \mathbb{Z} \times \mathbb{Z}, g\left( \begin{bmatrix} a & b \\ b & a \\ \end{bmatrix}\right)=(a-b,a+b) $, which fails to be a surjection, even though it is injective homomorphism.

Appre7
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2 Answers2

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You can write every element in $R$ as $aI+bJ$ with $$ I=\begin{pmatrix} 1 & 0 \cr 0 & 1 \end{pmatrix},\; J=\begin{pmatrix} 0 & 1 \cr 1 & 0 \end{pmatrix}. $$ We have $I^2=I, J^2=I$ and $IJ=JI=J$. The ring is isomorphic to $$ R\cong \Bbb Z[X]/(X^2-1) $$ Indeed, we can define a unique homomorphism $$ f\colon\mathbb{Z}[X]\to R $$ by $f(X)=J$. Then $f$ is surjective and its kernel is the ideal generated by $X^2-1$.

Note that this ring is not isomorphic to $\Bbb Z\times \Bbb Z$, see here:

$\mathbb{Z}[X]/(X^2-1)$ is not isomorphic with $\mathbb{Z}\times \mathbb{Z}$

See also the following post

Ring-isomorphism $\mathbb{Z}[X]/(X^2 - 1) \cong R $

Dietrich Burde
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The ring $\mathbb{Z}\times\mathbb{Z}$ has two nonzero orthogonal idempotents (elements $e=(1,0)$, $f=(0,1)$ which satisfy $e^2=e$, $f^2=f$, and $ef=(0,0)=fe$). But the ring of symmetric $2\times2$ integer matrices only has one nonzero idempotent, the identity$I_2$. (Check this by writing $A^2=A$ and solving $b=0$, $a=-1$.) So the rings cannot be isomorphic. But your $(a+b,a-b)$ map shows the matrix ring is isomorphic to the subring of $\mathbb{Z}\times\mathbb{Z}$ of pairs of integers with the same parity.

coiso
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