local ring

In fact, the fields are exactly the regular local rings of dimension 0.

In particular, it is a regular local ring of dimension n.

In particular, if is a prime power, then is a local ring.

A regular local ring is an example of such a ring.

The local ring in question is then the carrier of the formal moduli.

In the local ring case, it does matter.

Local rings play a major role in valuation theory.

The following is an example (due to Bass) of a local ring which is right but not left perfect.

It had such a nice local ring to it.

In other words, it suffices to consider local rings.

Let M be a finitely generated module over a local ring.

Then R is a finite local ring which is not principal.

Local rings arise in geometry as the germs of functions at a point.

At a foundational level we must use the second style of definition, to describe what a local ring means in a category.

For an element x of the local ring R, the following are equivalent:

A projective module over a local ring is free.

Local algebra - a term sometimes applied to the theory of local rings.

Over a local ring, one can say more about module epimorphisms:

Every field is a regular local ring.

A is a finite product of commutative Artinian local rings.

All complete Noetherian local rings, and in particular all fields, are excellent.

A ring with a unique maximal left ideal is a local ring.

A local ring announcer was used instead of the usual Bruce Buffer.

Other local rings have unique generic and special points, but a more complicated spectrum, since they represent general dimensions.

A property is said to be local if it can be detected from the local rings.