= Injectivity of a module map is detected at maximal localizations
{title2=$f\text{ injective}\iff f_{\mathfrak m}\text{ injective for every maximal }\mathfrak m$}
For a <module homomorphism> $f:M\to N$ over a <commutative ring> $A$, $f$ is <injective> if and only if every $f_{\mathfrak m}$ is <injective>, with $\mathfrak m$ ranging over <maximal ideals>. In a <localization of a module>, $x/s$ maps to zero exactly when some denominator $t$ satisfies $tf(x)=0$. If $f$ is <injective> this gives $tx=0$, so the original fraction is zero. In general it identifies $\ker(f_{\mathfrak m})$ with $(\ker f)_{\mathfrak m}$. A nonzero element of $\ker f$ has a proper <annihilator>, contained in some <maximal ideal>; it remains nonzero in that <localization>. Thus vanishing of all localized kernels forces $\ker f=0$. No finiteness or <Noetherian> assumption is required.
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