**Definition 1.** The **Jacobson radical** of a commutative ring is defined as the intersection of all maximal ideals of

**Proposition 2.** Let be a ring and be the set of all nonunits of Then the following are equivalent.

- is an ideal.
- has a unique maximal ideal.

*Proof.* If is an ideal, then it is maximal since any bigger ideal would contain a unit, and hence equal the whole ring. Uniqueness follows by adjoining nonunits. Conversely, note that the product of a unit and a nonunit, as well as the product of two nonunits, must be a nonunit. If has a unique maximal ideal, then note that if and is a unit, then there is a unit such that Hence the ideals generated by and are relatively prime. Hence no maximal ideal can contain both and so uniqueness is violated.

**Definition 3.** is a **local ring** if it satisfies the above conditions.

If is a local ring, then of course is the Jacobson radical of Note that if is a prime ideal in then its complement is a multiplicative subset.

**Definition 4.** We define We may call the **localization of** **at** We define the **globalization of** as the ring

where is a maximal ideal in

So if is a local ring, then its globalization is just since the one factor would invert already invertible elements.

**Proposition 5.** The set of maximal ideals of is finite iff is a finite direct sum of fields (where is the Jacobson radical of ).

*Proof.* is a field for each maximal ideal in Moreover if then maps onto via the canonical mapping. The kernel is clearly so we have the quotient as a finite sum of fields. Conversely, if is a finite direct sum of fields, then it has a finite number of ideals, and hence a finite number of maximal ideals. Since each maximal ideal in contains these are just the preimages of the maximal ideals of under the map of which there are finitely many.

**Definition 6.** A ring is **semilocal** if it is a finite direct sum of local rings.

Hence if then is semilocal, as fields are local. Now suppose is a semilocal ring. So each is local, with a maximal ideal We have a canonical epimorphism

whose kernel is Hence if is semilocal, then is semilocal (and in particular semisimple (as an -module), which is the traditional definition of being semilocal).

**Proposition 7.** Let be a finite set of prime ideals of Let us define

Then if is semilocal (i.e. is semilocal in the traditional sense), and has maximal ideals where are the maximal elements of

*Proof.* A maximal ideal in must be contained in otherwise it contains a unit. Hence it must be contained in one of the and hence it must be a So are the maximal ideals of Hence by Proposition 5 is semilocal.

[1] Bourbaki, N. *Commutative Algebra, Ch 1-7*. Springer-Verlag. 1989.