Local ring

A ring $A$ is said to be a local ring if it has a unique maximal ideal $m$. It is said to be semi-local if it has finitely many maximal ideals.

The localisation of a commutative integral domain at a non-zero prime ideal is a local ring.

Properties

In a local ring the unit group is the complement of the maximal ideal.

Complete local ring

A local ring A is complete if the intersection $\bigcap _{n}m^{n}=\{0\}$ and A is complete with respect to the uniformity defined by the cosets of the powers of m.

References

• Serge Lang (1993). Algebra, 3rd ed. Addison-Wesley, 100,206-207. ISBN 0-201-55540-9.