If R is a ring with unit element 1 and f is a homomorphism of R into an integral domain R' such that I(f) /= R. Prove that f(1) is the unit element of R'.

 

for 1, a in R, 1.a = a.1 = a. Then f(1.a) = f(1)f(a) = f(a)f(1)= f(a) ( since R' is commutative) so we see that f(1) is the unit element of R'. Please check my proof and let me know of any mistakes I made. thanks in advance.

Actually it´s quite ok. Just two remarks:

 

- Might be a good idea to explicitely mention that there is only one unit in a ring. Because you just show that f(1) is a unit. Ok, that´s nitpicking perhaps.

 

- Your remark "since R' is commutative" is a bit disturbing. Simple reason: You said that nowhere before. Also, this proof is usually made for non-commutative rings. So I´d guess you´ve got a flaw there. The proof which does not use commutativity is quite similar to your attempt, though.

Avoid saying unit when you mean identity.

I´m not a native english speaker so I don´t really know the english term. By "unit" I meant the one-element and I used the term because Meital used it. Well possible that it´s called identity (would indeed even make more sense).

A unit in a ring is *any* element that has a multiplicative inverse (ie divides 1). This is a very unfortunate thing really given the other meanings of unit/unital/unitary.