1. ## Homomorphisms with identity

Give an example of a homomorphism f:R->S such that R has an identity, but S does not. Does this contradict the fact that if R is a ring with identity and f is surjective, then S is a ring with identity and f(1R)=1S?

Give an example of a homomorphism f:R->S such that S has an identity, but R does not.

I am having trouble finding examples.

Give an example of a homomorphism f:R->S such that R has an identity, but S does not. Does this contradict the fact that if R is a ring with identity and f is surjective, then S is a ring with identity and f(1R)=1S?

Give an example of a homomorphism f:R->S such that S has an identity, but R does not.

I am having trouble finding examples.