Extension and contraction of ideals
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In commutative algebra, the extension and contraction of ideals are operations performed on sets of ideals.
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[edit] 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,
[edit] Contraction of an ideal
If 
is an ideal of B, then 
is always an ideal of A, called the contraction 
of to A.
[edit] 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.
See also: Splitting of prime ideals in Galois extensions
[edit] Reference
- Atiyah, M. F., and I. G. MacDonald, Introduction to Commutative Algebra, Perseus Books, 1969, ISBN 0-201-00361-9
