Extension and Contraction
Let be a ring homomorphism. Let . The extension of (with respect to ) is . Then contraction of (with respect to ) is +++i, which is an ideal in +.
Note that if is prime than so is , though the same is not always true for extensions. For example, take the identity map . Then for any prime , is prime in but which is not prime in .
In general there is no simple relationship between the prime ideals of and . For example consider the identity map (see notes on <a href="../numberfield"> number fields</a>).
Other properties of extension and contraction:
-
-
-
Let be the set of contracted ideals in and be the set of extended ideals in . Then , , and for all defines a bijection whose inverse is for all .
Let and . Then
-
,
-
,
-
,
-
,
-
,
The set is closed under sum and product while the set is closed under intersection, forming ideal quotients and taking radicals.