I read the following example in a book.
For any modules the zero map $0:A \to B$ given by $a \mapsto 0$ (for $a \in A$) is a module homomorphism. Every homomorphism of abelian groups is a $Z$-modules homomorphism.
If $R$ is a ring, the map $R[x] \to R[x]$ given by $f \mapsto xf$ (for example, $(x^2 + 1) \mapsto x(x^2 + 1)$) is an $R$-module homomorphism, but not a ring homomorphism.
But I did not understand:
1 - Why should every homomorphism of abelian groups be a $Z$-module homomorphism? Could anyone explain this for me please?
2 - Also I do not understand why the given map is an $R$-module homomorphism but is not a ring homomorphism. Could anyone clarify this for me please?