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.