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