Let $\displaystyle \phi : R \rightarrow R' $ be a ring homomorphism and let N be an ideal of R

(a) Show that $\displaystyle \phi [N] $ is an ideal of $\displaystyle \phi [R] $

(b) Give an example to show that $\displaystyle \phi [N] $ need not be an ideal of R'

(c) Let N' be an ideal either of $\displaystyle \phi [R] $ or of R'. SHow that $\displaystyle \phi^{-1} [N'] $ is an ideal of R.