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I got a problems as follow

Let $S = \left\{\begin{bmatrix} a & 0 \\ 0 & a \\ \end{bmatrix} | a \in R\right\}$, where $R$ is the set of real numbers. Then $S$ is a ring under matrix addition and multiplication. Prove that $R$ is isomorphic to $S$.

What is the key to prove it? By definition of ring?
But I have no idea how to connect the characteristic of ring to Real number.

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up vote 5 down vote accepted

Take the first map that occurs to you (it should be the right one!) from $\mathbf{R}$ to $S$, and use the definition of isomorphism to verify that it is indeed an isomorphism of rings.

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Once the isomorphism is verified, it automatically guarantees that the operations given in $S$ has the same properties as in $\Bbb R$, hence the ring axioms hold. – Berci Jan 29 '13 at 15:25
@Berci True, but for somebody grappling with a question like this for the first time, I would suggest first verifying that $S$ is a ring using the definition. Then I would suggest following Andreas's suggestion for the isomorphism. – Brett Frankel Jan 29 '13 at 16:00
Thanks! @BrettFrankel I followed your advice and indeed it's much more clear! – lucasKoFromTW Jan 30 '13 at 13:52
@Andreas I take your steps, and I did my proof! Thanks a lot! – lucasKoFromTW Jan 30 '13 at 13:52

Hint: Identify $a$ with \begin{bmatrix} a & 0 \\ 0 & a \\ \end{bmatrix} for each $a$ in $R$.

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