Going up and going down

The sets of Cohen- Seidenberg, named after Irvin Cohen and Abraham Seidenberg, two sets from the mathematical field of commutative algebra. They are also known as Going up and going down and deal with prime ideal chains in ring extensions.

Situation

Let be a ring extension of two commutative rings with the same identity element. Are and prime ideals, one says lies about if.

Is a prime ideal, a prime ideal in and is above. Is a Primidealkette with real inclusions in, a Primidealkette with real inclusions is. Here we turn to the question whether, conversely, can "lift" Primidealketten in to such after, so that the prime ideals in the chain are higher than those of the given chain. For this you have to make sure that lie over the prime ideals in prime ideals always made ​​first of all.

Looking around the ring expansion and is a prime, then the principal ideal generated is a prime ideal and there is no prime ideal that is above. Is it at all but a ring expansion, as one can show that is above every prime ideal of always a prime ideal of.

Thus, if an entire ring expansion and a Primidealkette in, so you can find for each prime ideal lying over a. It raises the question whether the well can be chosen such that they form an ascending chain. Just answer this question, the sets of Cohen- Seidenberg.

Going up

It is a whole ring expansion, a Primidealkette in and the prime ideal lying above:

Then there is the prime ideals lying, which form an ascending chain:

Going down

If you start in the situation of going up - set instead of a prime ideal lying with a lying, so you have to provide additional requirements for a similar statement:

It is a whole ring expansion of integrity rings with normal, a Primidealkette in and the prime ideal is lying about:

Then there is the prime ideals lying, which form an ascending chain:

Importance

Primidealketten play an important role in calculating the dimensions of a ring. From the going up theorem results immediately for a whole ring expansion. Going down The set can be used to

To show where the polynomial ring in indeterminates over the body.