Let R and S be non-zero commutative rings with multiplicative identities 1R 1S, respectively. Let f: R → S be a ring homomorphism with f(1R) = 1S. Which of the following statements are true?
1
If f(a) is a unit in S for every non-zero element a ∈ R, then S is a field
2
If f(a) is a unit in S for every non-zero element a ∈ R, then f(R) is a field
3
If R is a field, then f(a) is a unit in S for every non-zero element a ∈ R
4
If a is a unit in R, then f(a) is a unit in S