Extension and contraction of ideals
Extension of an ideal
Let A and B be two commutative rings with unity, and let f : A → B be a (unital) ring homomorphism. If is an ideal in A, then need not be an ideal in B (e.g. take f to be the inclusion of the ring of integers Z into the field of rationals Q). The extension of in B is defined to be the ideal in B generated by . Explicitly,
Contraction of an ideal
If is an ideal of B, then is always an ideal of A, called the contraction of to A.
Assuming f : A → B is a unital ring homomorphism, is an ideal in A, is an ideal in B, then:
- is prime in B is prime in A.
- It is false, in general, that being prime (or maximal) in A implies that is prime (or maximal) in B. Many classic examples of this stem from algebraic number theory. For example, embedding . In , the element 2 factors as where (one can show) neither of are units in B. So is not prime in B (and therefore not maximal, as well). Indeed, shows that , , and therefore .
On the other hand, if f is surjective and then:
- and .
- is a prime ideal in A is a prime ideal in B.
- is a maximal ideal in A is a maximal ideal in B.
Extension of prime ideals in number theory
Let K be a field extension of L, and let B and A be the rings of integers of K and L, respectively. Then B is an integral extension of A, and we let f be the inclusion map from A to B. The behaviour of a prime ideal of A under extension is one of the central problems of algebraic number theory.