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In the paper How to construct huge chains of prime ideals in power series rings by B. Kang and P. Toan the Krull dimension of a commutative ring with $1$ is defined as follows:

Let $R$ be a commutative ring and $\mathfrak{C}$ be an arbitrary chain of prime ideals in $R$. Then length of $\mathfrak{C}$ is defined by $\left\vert{\mathfrak{C}}\right\vert-1$. The Krull dimension of $R$ is the largest cardinal number $\alpha$ (if any) such that there exists a chain of prime ideals in $R$ whose length is equal to $\alpha$. We write $\dim(R)\geq \alpha$ if there is a chain of prime ideals in $R$ whose length is $\geq \alpha$.

Note that this agrees with the usual definition of dimension when $\dim(R)$ is finite. However, there is no guarantee that the dimension of a ring exists in general. In the above paper, there are classes of rings with uncountable chains of prime ideals. Assuming that the Continuum hypothesis is true, there are a few examples where the Krull dimension is determined. This is done by noting that $\dim(R) \leq \left\vert{2^R}\right\vert$.

Does the dimension of a commutative ring exist in general? Is the existence of dimension for all commutative rings equivalent to the generalized continuum hypothesis?

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    $\begingroup$ In the noetherian case the natural generalization of Krull dimension is by ordinals rather than cardinals. Namely, define inductively $\ell(A)=\sup(\ell(A/I)+1)$ where $I$ ranges over nonzero ideals of $A$. Then the ordinal Krull dimension of $A$ is the unique $\alpha$ such that $\omega^\alpha\le\ell(A)<\omega^{\alpha+1}$. $\endgroup$ – YCor Jun 4 '15 at 20:44
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Here is one stupid obstruction.

Take any limit cardinal $\alpha = \bigvee_{i \in I} \alpha_i$, $\alpha_i < \alpha$, such that there is a ring $R_i$ with chains of any length $< \alpha_i$ but not of length $\alpha_i$. Then the ring $R$ defined as the unitalizatioon of $\bigoplus_i R_i$ doesn't have a dimension.

Indeed, any prime ideal is necessarily supported on some $i$, i.e. is pulled back from one of $R_i$, so every chain of prime ideals of $R$ is pulled back from a chain in some $R_i$. These don't have a maximal cardinality.

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    $\begingroup$ In any case, there is always a smallest cardinality $\alpha$, such that there are no chains of length $\ge \alpha$, which can serve as an alternative notion of dimension. $\endgroup$ – Alexander Shamov Jun 14 '15 at 5:40
  • $\begingroup$ Cardinals can be constructed as a sub-class of ordinals, so they are well-ordered. $\endgroup$ – Alexander Shamov Jun 15 '15 at 11:37

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