degree of a map in terms of fundamental classes confusion Hatcher makes the following definition in exercise 7 on page 258. 
For a map $f: M \rightarrow N $ between connected, closed, orientable $n$-manifolds with fundamental classes $[M]$ and $[N]$, the degree of $f$ is defined to be the integer $d$ such that $f_{*}[M]=d[N]$ (so the sign of the degree depends on the choice of fundamental classes).

I'm confused as to why $f_{*}[M]$ has be of the form $d[N]$ for arbitrary coefficient ring $R$. I know that $[M]$ will be a generator in the group that it lies in, namely $H_{n}(M;R) \approx R$ and similarly $[N]$ is a generator for $H_{n}(N,R) \approx R$. I showed that for any isomorphism $f:R\rightarrow S$ a generator of $R$ must map to a generator of $S$, where my definition of generator for $R$ is an element $u$ such that $Ru=R$. But with this I still can't see why $f_{*}[M]=d[N]$. 


For the definition of degree, you should consider homology with integer coefficients. Now, $$f_*[M]\in H_n(N,\mathbb{Z})=\mathbb{Z}\langle[N]\rangle,$$and so, there is a unique $d\in\mathbb{Z}$ such that $$f_*[M]=d[N].$$ 
