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

It had such a nice local ring to it.

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

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

In other words, it suffices to consider local rings.

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

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

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