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

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