The Stacks project

Definition 10.17.1. A local ring is a ring with exactly one maximal ideal. The maximal ideal is often denoted $\mathfrak m_ R$ in this case. We often say “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.

Comments (0)

There are also:

  • 2 comment(s) on Section 10.17: Local rings

Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 07BI. Beware of the difference between the letter 'O' and the digit '0'.