Krull's principal ideal theorem

This article gives the statement and possibly, proof, of an implication relation between two commutative unital ring properties. That is, it states that every commutative unital ring satisfying the first commutative unital ring property must also satisfy the second commutative unital ring property
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Statement

Symbolic statement

Let $R$ be a Noetherian and $x\in R$. Let $P$ be a minimal element among prime ideals containing $x$. Then, the codimension of $P$ is at most 1.

Property-theoretic statement

The property of commutative unital rings of being a Noetherian ring is stronger than the property of being a ring satisfying PIT.

Generalizations

• Krull's height theorem: This is often also called the final version of the principal ideal theorem.
• Determinantal ideal theorem: This generalizes the principal ideal theorem to the ideal generated by the determinants of minors of a matrix