TITLE: Can a ring without a unit element have a subring with a unit element?
QUESTION [19 upvotes]: A ring need not have a unit element, e.g. even numbers. 

Can someone give me an example of a ring which doesn't have a unit element but it's subring does have a unit element.

REPLY [3 votes]: Here is another example based on matrices. Let $K$ be a field. Consider the set $R$ of matrices of type 
$$
\begin{pmatrix}
a & b \\
0 & 0
\end{pmatrix} \subset K^{2,2}.
$$
Supply $R$ with the usual matrix addition and multiplication. Then $R$ is a ring without unit element. In fact
$$
\begin{pmatrix}
1 & b \\
0 & 0
\end{pmatrix}
$$
is a left-sided unit for all $b$, but there is no right unit.
However, the set of matrices of type
$$
\begin{pmatrix}
a & 0 \\
0 & 0
\end{pmatrix} \subset K^{2,2}.
$$
is a subring of $R$, which is even a field.