Extension and contraction of ideals

In commutative algebra, the extension and contraction of ideals are operations performed on sets of ideals.

Extension of an ideal

Let A and B be two commutative rings with unity, and let f : AB 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\}

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.

Properties

Assuming f : AB is a unital ring homomorphism, \mathfrak{a} is an ideal in A, \mathfrak{b} is an ideal in B, then:

It is false, in general, that \mathfrak{a} being prime (or maximal) in A implies that \mathfrak{a}^e is prime (or maximal) in B. Many classic examples of this stem from algebraic number theory. For example, embedding \mathbb{Z} \to \mathbb{Z}\left\lbrack i \right\rbrack. In B = \mathbb{Z}\left\lbrack i \right\rbrack, the element 2 factors as 2 = (1 + i)(1 - i) where (one can show) neither of 1 + i, 1 - i are units in B. So (2)^e is not prime in B (and therefore not maximal, as well). Indeed, (1 \pm i)^2 = \pm 2i shows that (1 + i) = ((1 - i) - (1 - i)^2), (1 - i) = ((1 + i) - (1 + i)^2), and therefore (2)^e = (1 + i)^2.

On the other hand, if f is surjective and  \mathfrak{a} \supseteq \mathop{\mathrm{ker}} f then:

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.

See also

References

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