Vipul: 1 revision

2008-05-12T16:22:41Z

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Vipul: New page: ==Statement== This result is sometimes called ''going up'' and sometimes ''lying over and going up''. It is a stronger version of lying over. Suppose $f:R \to S$ is an inj...

2008-02-02T20:09:39Z

New page: ==Statement== This result is sometimes called ''going up'' and sometimes ''lying over and going up''. It is a stronger version of lying over. Suppose <math>f:R \to S</math> is an inj...

New page

==Statement==

This result is sometimes called ''going up'' and sometimes ''lying over and going up''. It is a stronger version of [[lying over]].

Suppose <math>f:R \to S</math> is an injective homomorphism of commutative unital rings, such that <math>S</math> is an [[integral extension]] of <math>R</math>. Suppose <math>P</math> is a [[prime ideal]] of <math>R</math>, and <math>Q_1</math> is an ideal of <math>S</math> such that <math>f^{-1}(Q_1) \subset P</math>. Then, there exists a [[prime ideal]] <math>Q</math> containing <math>Q_1</math>, such that <math>f^{-1}(Q) = P</math>.

==Proof==

This follows from [[lying over]], applied to the injective map <math>R/f^{-1}(Q_1) \to S/Q_1</math>.

Going up theorem - Revision history

Vipul: 1 revision

Vipul: New page: ==Statement== This result is sometimes called ''going up'' and sometimes ''lying over and going up''. It is a stronger version of lying over. Suppose f:R \to S is an inj...

Vipul: New page: ==Statement== This result is sometimes called ''going up'' and sometimes ''lying over and going up''. It is a stronger version of lying over. Suppose $f:R \to S$ is an inj...