Should Krull dimension be a cardinal? • A chain of length …

Should Krull dimension be a cardinal?

• A chain of length \( n \) in a ring \( A \) consists of a totally ordered finite set of prime ideals, denoted as \( \mathcal P_0 \varsubsetneq \mathcal P_1 \varsubsetneq \dots \varsubsetneq \mathcal P_n \).

• The Krull dimension \( \dim(A) \) of the ring \( A \) is defined as the supremum of the lengths of such chains of prime ideals.

• If the lengths of these chains are unbounded, the ring is classified as infinite dimensional, indicated by \( \dim(A) = \infty \), which can occur even in Noetherian rings.

• For infinite dimensional rings, one can examine arbitrary totally ordered subsets of prime ideals \( \Pi \subset Spec(A) \) and their cardinality \( card(\Pi) \).

• The supremum of all these cardinalities is known as the cardinal Krull dimension of the ring \( A \).

• An equality \( \dim(A) = \aleph \) offers a more precise measure of the infinite dimensionality of \( A \) than simply stating \( \dim(A) = \infty \).

• The author seeks known results regarding the cardinal Krull dimension, particularly for the ring of continuous functions \( \mathcal C(X) \) on a topological space \( X \).

• The author finds it challenging to determine the cardinal Krull dimension of \( \mathcal C(X) \), even for the specific case where \( X = \mathbb R \).

• The inquiry also includes variants related to rings of differentiable functions on manifolds and similar mathematical structures.