Vipul: New page: {{curing property implication| stronger = principal ideal ring| weaker = one-dimensional ring}} ==Statement== ===Verbal statement=== Any principal ideal ring is a [[one-dimensional ...

2009-02-08T23:43:32Z

New page: {{curing property implication| stronger = principal ideal ring| weaker = one-dimensional ring}} ==Statement== ===Verbal statement=== Any principal ideal ring is a [[one-dimensional ...

New page

{{curing property implication|
stronger = principal ideal ring|
weaker = one-dimensional ring}}

==Statement==

===Verbal statement===

Any [[principal ideal ring]] is a [[one-dimensional ring]]: its [[fact about::Krull dimension]] is at most one. In other words, it cannot have an ascending chain of [[fact about::prime ideal]]s of length more than one.

===Statement with symbols===

Suppose <math>R</math> is a [[principal ideal ring]]. Then, we cannot have a ''strictly ascending'' chain of prime ideals in <math>R</math> of the form:

<math>P_0 \subset P_1 \subset P_2</math>

==Related facts==

* [[PID implies one-dimensional]]
* [[Unique factorization and one-dimensional iff principal ideal]]
* [[Unique factorization implies every nonzero prime ideal contains a prime element]]
* [[Unique factorization and finite-dimensional implies every prime ideal is generated by a set of primes of size at most the dimension]]

==Proof==

'''Given''': A principal ideal ring <math>R</math>.

'''To prove''': There cannot be three prime ideals <math>P_0 \subset P_1 \subset P_2</math> with the containment strict.

'''Proof''': Suppose we have such prime ideals. Since <math>R</math> is a principal ideal ring, we can choose generators <math>p_1, p_2</math> of <math>P_1, P_2</math>. We then have <math>p_1 = q_{12}p_2</math> for some <math>q_{12} \in R</math>. Since <math>P_1</math> is prime and <math>p_2 \notin P_1</math>, we have <math>q_{12} \in P_1</math>. Thus, <math>q_{12} = p_1x</math> for some <math>x \in R</math>. This yields:

<math>p_1 = p_1xp_2</math>

which simplifies to:

<math>p_1(1 - xp_2) = 0</math>.

Since <math>0 \in P_0</math> and <math>P_0</math> is prime, we have that either <math>p_1 \in P_0</math> (not possible since the containment is strict) or <math>xp_2 = 1</math> (not possible since <math>P_2</math> is a proper ideal). Thus, we have the required contradiction.

Principal ideal ring implies one-dimensional - Revision history

Vipul: New page: {{curing property implication| stronger = principal ideal ring| weaker = one-dimensional ring}} ==Statement== ===Verbal statement=== Any principal ideal ring is a [[one-dimensional ...