R and S are rings.
Show that if is a homomorphism, and if a is a unit of S. Show, in fact, that for any unit of a of R.
Is this how I show this?
No. When you use the expression you have assumed it exists, but you have to first show that it exists. Also, in the first equality it looks like (trying to read around your type) you have assumed that , which is neither given.
Instead, just notice that . The conclusion follows immediately.
My book states the following:
f is a homomorphism if f satisfies the following properties:
(1)
(2)
(3)
If f satifies 1-3, then
(4)
(5)
So I didn't assume anything. I used what was given. It is stated that f is a homomorphism so it satisfies 1-3 which means 4 and 5 are also satisfied.