Ring Homomorphism: unit in R implies unit in R'

Not all rings are unitary, so the definition of ring homomorphism given on wikipedia is for a general ring. When a ring is unitary, most people require that a homomorphism map 1 to 1, but occasionally you will find authors who do not use this convention (in this case, 1 just maps to some unit in the ring).

 

It need not be true if the homomorphism does not map 1 to 1'.

For example, consider the map [itex]f : \mathbb{Z} \rightarrow M_{2 \times 2}(\mathbb{R})[/itex] defined by
[tex]f(n) = \left[\begin{matrix}
n & 0 \\
0 & 0
\end{matrix}\right][/tex]
This is a ring homomorphism that does not map 1 to 1', and clearly the image contains no units.