Let HRS be a homomorphism from the ring R to the ring S Prov

Let H:RS be a homomorphism from the ring R to the ring S. Prove that if J is an ideal of S, then the set I = {b R | H(b) = c for some element c J} is an ideal of R.

Solution

Let r is in R

And b in I => H(b) = c for some c in J

Now as r and b are in R and so R being a ring rb is in R

H(rb) = H(r)H(b) (Because H is a ring homomorphism)

=> H(rb) = H(r)c (because H(b) = c)

As J is ideal of S and H(r) is in S so H(r)c is in J

=> H(rb) = H(r)c is in J

Hence we get that rb is in I (Using the definition of I)

So I is an ideal of R