Radicals of ideals
Let be a ring, an ideal. The radical of , denoted , is
A radical ideal is an ideal such that .
Let be a ring, an ideal. Then
For , note that . For , assume for no . Form the quotient ring . By theorem 12.3, pick a prime ideal disjoint from the multiplicatively closed set . Form the ideal . is a prime ideal which contains and does not intersect . Hence is not in the right hand side.
The radical of an ideal is an ideal.
Intersection of ideals is an ideal.
Let be a ring, . The Jacobson radical of is defined as thus:
Let be a ring, . is a radical ideal.
Clearly, . Further from theorem 13.2; the last equality from .
The radicals of the zero ideal
Let be a ring. is an ideal. The nilradical of , written , is defined as
Note that by definition
the set of nilpotent elements.
Let be a ring. is an ideal. The Jacobson radical of , written , is defined as
We have .
If is a ring, is the set of units of .
Let be a ring, its Jacobson radical. Then
: Let , . Assume . Form the ideal ; by theorem 12.8 there exists maximal with , hence . If , then , contradiction.
: Assume for all and . Then there is a maximal ideal not containing . Hence and for a and an . Hence is not a unit.
Radicals and localisation
If is a ring, is a multiplicative subset, an ideal, set
the localisation of the ideal with respect to .
Let be a ring, an ideal, a multiplicatively closed subset. Then
Let , that is, . Then , , . There exists such that . Thus , whence and . Thus, .
Let . We may assume . Choose such that . Then , whence .
Strong Nakayama lemma
- Prove that whenever is a reduced ring, then the canonical homomorphism is injective.