 Local homomorphism of local rings (source code)

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= Local homomorphism of local rings
{title2=$\varphi^{-1}(\mathfrak n)=\mathfrak m$}

= Local homomorphism
{synonym}

A homomorphism $\varphi:(A,\mathfrak m)\to(B,\mathfrak n)$ of <local rings> is local if $\varphi(\mathfrak m)\subseteq\mathfrak n$, equivalently $\varphi^{-1}(\mathfrak n)=\mathfrak m$. The equivalence uses the fact that all elements outside $\mathfrak m$ are <units>. It induces an embedding of <residue fields> $A/\mathfrak m\to B/\mathfrak n$.

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