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.

I presume we are talking about rings here? "Homomorphism" is a pretty ambiguous term...

For the first question, take the ring of 2x2 real (or whatever) matrices, then this is embedded in the ring of 3x3 real matrices with bottom row zero in a natural way. The former has identity, the latter does not. The key is surjectivity! The image of the homomorphism has an identity, but the ring we are mapping into doesn't.

For the second question, take the ring of 3x3 real (or whatever) matrices with bottom row consisting of zeros. Can you think of a nice homomorphism from this back into this? Essentially, do the opposite from what I just did...