Definition 10.18.1. A *local ring* is a ring with exactly one maximal ideal. If $R$ is a local ring, then the maximal ideal is often denoted $\mathfrak m_ R$ and the field $R/\mathfrak m_ R$ is called the *residue field* of the local ring $R$. We often say “let $(R, \mathfrak m)$ be a local ring” or “let $(R, \mathfrak m, \kappa )$ be a local ring” to indicate that $R$ is local, $\mathfrak m$ is its unique maximal ideal and $\kappa = R/\mathfrak m$ is its residue field. A *local homomorphism of local rings* is a ring map $\varphi : R \to S$ such that $R$ and $S$ are local rings and such that $\varphi (\mathfrak m_ R) \subset \mathfrak m_ S$. If it is given that $R$ and $S$ are local rings, then the phrase “*local ring map $\varphi : R \to S$*” means that $\varphi $ is a local homomorphism of local rings.

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