= Finite generation clears localization kernels
{title2=$R_f\hookrightarrow R_P$}
For a <prime ideal> $P$, suppose the <kernel of a ring homomorphism> $K=\ker(R\to R_P)$ is a finitely generated <ideal>, say $(a_1,\ldots,a_t)$. Choose $s_i\notin P$ killing $a_i$ and let $f=\prod_i s_i$. Then $f\notin P$ and $fK=0$. If a fraction $a/f^n$ maps to zero in $R_P$, then $a\in K$, so $fa=0$ and the fraction was already zero in $R_f$. Thus the canonical <ring homomorphism> $R_f\to R_P$ is <injective>. The empty product $f=1$ handles $K=0$. Every <Noetherian ring> satisfies the finite generation hypothesis.
\Comment[[
Exact target: noninjective-surjective-shift-on-a-product-of-fields
Existing parent: Direct product of rings
Intended file: /home/ciro/bak/git/codex-wiki/commutative-algebra.bigb
]]
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