I have the following doubt:
Let $f:M\rightarrow N$ be a morphism of $R$-modules where all the elements of $\ker(f)$ are annihilated by $p$. ($R$ is a ring that contains the integers.)
Now if $M$ has an element $z$ which is annihilated by $q$, where $p$ and $q$ are coprime, I consider $T$ the submodule generated by $z$.
How can I see that the composition $$T\rightarrow M \rightarrow N$$
is injective?
If I take one element in the kernel of that composition, that element is in the kernel of $f$. Thus I have one element that is annihilated by $p$ and $q$.
Does it follows from the fact that
$$1\cdot z= (pt +ql)z=0\ ?$$