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.

1 Extension of an ideal
2 Contraction of an ideal
3 Extension of prime ideals in number theory
4 References

[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 $\mathfrak{a}$ is an ideal in A, then $f(\mathfrak{a})$ 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 $\mathfrak{a}^e$ of $\mathfrak{a}$ in B is defined to be the ideal in B generated by $f(\mathfrak{a})$ . Explicitly,

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

[edit] 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.

[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 $\mathfrak{a} = \mathfrak{p}$ of A under extension is one of the central problems of algebraic number theory.