Let R be a ring and f: M→N is an R-homomorphism. [closed]

Let R be a ring and f: M→N is a R homomorphism.

1. If f is monomorphism Annhilator(N)⊆ Annihilator(M).
2. If f is an isomorphism Annhilator(M)=Annihilator(N).

I need to prove these propositions, I started to prove them by the definition of monomorphism and isomorphism but couldn't connect with the annhilator. Also I used the if f is monomorphism Kerf=0 theorem too.

Solution 1:

Let $r \in \operatorname{Ann}(N)$. We need to show that $r \in \operatorname{Ann}(M)$, that is, that $r \cdot m = 0$ for all $m \in M$. Since $f$ is a monomorphism, it is enough to show that $f(r \cdot m )=0$. Can you go from here?

The second proposition is implied by the first (hint: consider the inverse of $f$).

Solution 2:

Hint on 1).

If $r$ annihilates $N$ then:

$f(rm)=rf(m)=0=f(0)$ for every $m\in M$.