Ring Homomorphism: unit in R implies unit in R'

[AcidRainLiTE]

I was just looking at wikipedia's article on ring homomorphisms (http://en.wikipedia.org/wiki/Ring_homomorphism) and I am a little confused.

If you look at the definition they give for ring homomorphism, they require only that addition and multiplication is preserved over the homomorphism (and not that it maps 1 to 1'). Then, under the 'Properties' section, the third bullet down claims:

If a has a multiplicative inverse in R, then f(a) has a multiplicative inverse in S and we have f(a⁻¹) = (f(a))⁻¹. Therefore, f induces a group homomorphism from the (multiplicative) group of units of R to the (multiplicative) group of units of S.

Don't you need to explicitly require that the homomorphism maps 1 to 1' in order for that statement to be true? Or is there someway to deduce this without specifying that requirement?

 

[jgens]

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).

 

[jbunniii]

Don't you need to explicitly require that the homomorphism maps 1 to 1' in order for that statement to be true? Or is there someway to deduce this without specifying that requirement?

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

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

 

[AcidRainLiTE]

Thanks for the quick response!