I've been looking at the extension of Krull dimension to non-commutative rings as defined, for example, in On the Krull-Dimension of Left Noetherian Left Matlis-Rings [Krause, Mathematische Zeitschrift Vol. 118, 1970].
It is shown there that for Noetherian commutative rings, this definition coincides with the usual definition in terms of prime ideals (there called the "classical" Krull dimension). It is mentioned on the Wikipedia page for Krull dimension that these dimensions can differ even in the commutative case when the ring is not Noetherian, but does not cite an example.
On the contrary, unless I'm misreading something, Corollary 5.5.4 of the book Dimensions of Ring Theory [Năstăsescu and van Oystaeyen, Mathematics and Its Applications, 1987] seems to say that these dimensions agree for commutative rings even without the Noetherian hypothesis.
Then my question is as follows:
Is there an example of a (non-Noetherian) commutative ring for which the classical Krull dimension is not equal to the generalisation defined in terms of the deviation of the poset of its submodules?
A reference to such an example would be very welcome, and any help would be much appreciated.
Update:
As suggested in the comments, I'll include a statement of the generalised definition. This is taken from the paper of Krause mentioned above:
Let $R$ be a ring and $M$ a left $R$-module. We denote by $\Gamma(M)$ the set of all pairs $(K,N)$ of $M$ with $N\subseteq K$. Let
$$\Gamma_0(M)=\{(K,N)\in\Gamma(M):\text{$K/N$ is Artinian}\}$$
and for ordinals $\alpha>0$, set
$$\begin{align*}
\Gamma_\alpha(M)=\{(K,N)\in\Gamma(M)&:\text{$K\supseteq K_1\supseteq\cdots K_i\supseteq K_{i+1}\supseteq\cdots\supseteq N$}\\
&\quad\text{implies $(K_i,K_{i+1})\in\bigcup_{\beta<\alpha}\Gamma_\beta(M)$ for almost all $i$}\}
\end{align*}$$
If there exists some ordinal $\alpha$ such that $\Gamma_\alpha(M)=\Gamma(M)$, then we say that the smallest such ordinal is the Krull dimension of M.
I'm particularly interested in the case where $\alpha$ is finite, since the paper Sur la dimension des anneaux et ensembles ordonnés [Rentschler and Gabriel, Comptes Rendus de l'Académie des sciences Série A 265, 1967], whose definition Krause generalises to possibly infinite ordinals, still includes the Noetherian hypothesis for equality with the classical Krull dimension.
Best Answer
The short answer is "there is no such example".
$\DeclareMathOperator{kdim}{K\,dim}$ $\DeclareMathOperator{clkdim}{cl\,K\,dim}$ The first place I would look for a reference is Krull Dimension by Gordon and Robson (Memoirs of the AMS 133). For a more recent but more Noetherian-oriented exposition, try Chapter 6 of Noncommutative Noetherian Rings by McConnell and Robson (AMS Graduate Studies in Mathematics 30).
Let's denote the Krull dimension defined via deviation of posets (the one the OP defined, which I'll call the noncommutative Krull dimension) by $\kdim R$. There are multiple notions of Krull dimension defined in terms of prime ideals (classical Krull dimension), depending on how one wants to deal with infinities. These definitions all agree when the dimension is finite. We'll denote the Gabriel-Krause version by $\clkdim R$. Every ring having noncommutative Krull dimension has acc on prime ideals and I think the classical Krull dimensions exist if and only if the ring has acc on prime ideals.
For all rings $R$ having noncommutative Krull dimension (even infinite), $\clkdim R\le \kdim R$: this is Proposition 7.9 in Gordon & Robson. For all commutative rings $R$ having noncommutative Krull dimension (even infinite), $\clkdim R=\kdim R$: this is Corollary 8.14 in Gordon & Robson. The last equality is even true if we relax the commutative hypothesis to right or left fully bounded (Theorem 8.12).