Past exam of the mathematics course of the University of Cambridge / 2004 / iii / Paper 1 / 2 / ii / Solution

Set $C=N/\phi (M)$. The natural map $N_P\to C_P$ is onto. Its kernel is the image of ${\phi }_P$: a fraction $n/s$ maps to zero precisely when $tn\in \phi (M)$ for some $t\notin P$. If $tn=\phi (m)$, then $n/s={\phi }_P(m/(ts))$. This proves the localized cokernel identity directly:Thus ${\phi }_P$ is a surjection for every $P$ exactly when $C_P=0$ for every $P$. Part (i) makes this equivalent to $C=0$, or to $\phi$ being a surjection. Hence surjectivity is detected at all prime localizations, and again maximal ideals alone would suffice.