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Suppose we are given an arbitrary ring $R$. Then the set $M_n(R)$ of all square matrices with elements from $R$, together with usual matrix addition and multiplication forms a ring. If R is a unitary ring then $M_n(R)$ too.

My question seems to be very trivial but is it possible that $M_n(R)$ has unity while $R$ hasn't?

Thanks in advance.

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Let $$ \begin{bmatrix}a & b\\c& d\end{bmatrix} $$ be the identity in $M_2(R)$. Then, for any $r$ in $R$, we have $$ \begin{bmatrix}r & 0\\0& 0\end{bmatrix}= \begin{bmatrix}r & 0\\0& 0\end{bmatrix} \begin{bmatrix}a & b\\c& d\end{bmatrix}= \begin{bmatrix}ra & rb\\0& 0\end{bmatrix} $$ so $ra=r$, for all $r\in R$, and $a$ is a right unity in $R$. Similarly $$ \begin{bmatrix}r & 0\\0& 0\end{bmatrix}= \begin{bmatrix}a & b\\c& d\end{bmatrix} \begin{bmatrix}r & 0\\0& 0\end{bmatrix}= \begin{bmatrix}ar & 0\\cr& 0\end{bmatrix} $$ so $a$ is also a left unity.

The example generalizes easily to any matrix size.

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  • $\begingroup$ Yes, it's really very simple. Thank you very much! $\endgroup$ – Igor May 25 '13 at 11:19

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