Definition 2

Let $\mathbf{A}, \mathbf{B}$ be two local rings and $m, n$ their respective maximal ideals. A homomorphism $u : \mathbf{A} \to \mathbf{B}$ is called local if $u(m) \subset n$.

This amounts to saying that $\bar{u}^{-1}(n) = m$, for $\bar{u}^{-1}(n)$ is then an ideal containing $m$ and not containing 1 and hence equal to $m$. Taking quotients, we then derive canonically from $u$ an injective homomorphism $\mathbf{A}/m \to \mathbf{B}/n$ from the residue field of $\mathbf{A}$ to that of $\mathbf{B}$.

Examples (1) A field is a local ring. A ring which is reduced to 0 is not a local ring. (2) Let $\mathbf{A}$ be a local ring and $k$ its residue field. The ring of formal power series $B = \mathbf{A}[[X_1, \ldots, X_n]]$ is a local ring, for the non-invertible elements of $B$ are the formal power series whose constant terms are not invertible in $\mathbf{A}$ (Algebra, Chapter IV, §5, no. 6, Proposition 4). The canonical injection of $\mathbf{A}$ into $B$ is a local homomorphism and the corresponding injection of residue fields is an isomorphism. (3) Let $b$ be an ideal of a ring $\mathbf{A}$ which is only contained in a single maximal ideal $m$; then $A/b$ is a local ring with maximal ideal $m/b$ and residue field canonically isomorphic to $A/m$. This applies in particular to the case $b = m^k$, $m$ being any maximal ideal of $A$ ($§ 1$, no. 1, Corollary to Proposition 1). If $A$ itself is a local ring with maximal ideal $m$, then for every ideal $b \neq A$ of $A$, $A/b$ is a local ring, the canonical homomorphism $A \to A/b$ a local homomorphism and the corresponding homomorphism of residue fields an isomorphism.

(4) Let $X$ be a topological space, $x_0$ a point of $X$ and $A$ the ring of germs at the point $x_0$ of real-valued functions continuous in a neighbourhood of $x_0$ (General Topology, Chapter I, § 6, no. 10). Clearly, for the germ at $x_0$ of a continuous function $f$ to be invertible in $A$, it is necessary and sufficient that $f(x_0) \neq 0$, as that implies that $f(x) \neq 0$ in a neighbourhood of $x_0$. The ring $A$ is therefore a local ring whose maximal ideal $m$ is the set of germs of functions which are zero at $x_0$; taking quotients, the mapping $g \mapsto g(x_0)$ of $A$ to $\mathbf{R}$ gives an isomorphism of the residue field $A/m$ onto $\mathbf{R}$.