Let f : R → S be a homomorphism of rings and let K = {r element of R : f(r) = 0_s}

A) Prove that if f is one-to-one, then K = {0_R}
B) Prove that if K = {0_R}, then f is one-to-one.

Can someone help please, and 10 points fast for you
Thanks

a)
We have f(0_R) = 0_S, as is straightforward to show.
Therefore if r ∈ K, i.e. f(r) = 0_S, then f(r) = f(0_R) and so by injectivity
of f it follows that r = 0_R.


b)
f(r) = f(r')
⇒ 0_S = f(r) - f(r')
⇒ 0_S = f(r - r')
⇒ r - r' = 0_R (since K = {0_R})
⇒ r = r'

→ f is injective.