Extension and contraction of ideals

In commutative algebra, the extension and contraction of ideals are operations performed on sets of ideal (ring theory)s.

= Extension of an ideal =

Let A and B be two of the ring of integers Z into the field of rationals Q). The extension mathfrak{a}^e of mathfrak{a} in B is defined to be the ideal in B generated by f(mathfrak{a}). Explicitly,

:mathfrak{a}^e = Big{ sum y_if(x_i) : x_i in mathfrak{a}, y_i in B Big}

= Contraction of an ideal =

If mathfrak{b} is an ideal of B , then f^{-1}(mathfrak{b}) is always an ideal of A , called the contraction mathfrak{b}^c of mathfrak{b} to A .

= Extension of prime ideals in number theory =

Let K be a field extension of L , and let B and A be the ring 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 mathfrak{a} = mathfrak{p} of A under extension is one of the central problems of algebraic number theory.

See also: Splitting of prime ideals in Galois extensions

= Reference =

*Atiyah, M. F., and I. G. MacDonald, Introduction to Commutative Algebra , Perseus Books, 1969, ISBN 0-201-00361-9