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" : "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,

:$mathfrak\{a\}^e\; =\; Big\{\; sum\; y\_if(x\_i)\; :\; x\_i\; in\; mathfrak\{a\},\; y\_i\; in\; B\; Big\}$

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 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: Splitting of prime ideals in Galois extensions

References

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