Let $R$ and $S$ be two commutative rings with unity and $f : R → S$ be a ring homomorphism.

Question

Let $R$ and $S$ be two commutative rings with unity and $f : R → S$ be a ring homomorphism.

Then which of the following statement(s) is/are true ?

$a.$ Image of an ideal of R is always an ideal in S.

$b$. Image of an ideal of R is an ideal of S if f is injective.

$c$. Image of an ideal of R is an ideal of S if f is surjective.

$d.$ Image of an ideal of R is an ideal of S if S is a field.

if i take $R = S = Z_2$ then option b ), c) and d) are corrects...

im confused about option 1)..

Any Hints/solution....

Thanks in advance