Let f: R --> R' be a ring homomorphism and let N be an ideal of R.
Show that f(N) is an ideal of f(R) and also show that f(N) doesn't need to be an ideal of R'
To show is an ideal basically what you need to show is that for all and . Now since it means for some and for some . Therefore, since for is an ideal, similarly . Thus, .
Consider defined by (evaluating polynomials at ).also show that f(N) doesn't need to be an ideal of R'
Let then .
This is not an ideal of since the complex numbers are a field.