# Unity in the rings of matrices

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.

## 1 Answer

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.

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