From the introduction: In this paper, we consider the Krull-Gabriel dimension, \(KG\), of a serial ring. In particular, we prove that for a serial ring \(R\), \(KG(R)\) exists if and only if the lattice of right (left) ideals of \(R\) has Krull dimension (in the sense of Gabriel and Rentschler). Moreover, if the Krull dimension of \(R\) is equal to \(\alpha\), then \(KG(R)\) does not exceed \(\alpha\oplus\alpha\), and we show that it is precisely \(\alpha\oplus\alpha\) for some classes of serial rings. Here \(\alpha\oplus\alpha\) stands for the Cantor sum of \(\alpha\) and \(\alpha\).…
All the results of this paper are tightly connected with the calculation of the Cantor-Bendixson rank of the Ziegler spectrum over a serial ring \(R\), \(CB(Zg_R)\), made by Puninski (1999) and Reynders (1999). In fact, we will prove that the so-called isolation property holds true over any serial ring \(R\): every isolated point in the Ziegler spectrum of any theory of \(R\)-modules is isolated by a minimal pair. An immediate consequence of this result is that the following invariants of a serial ring \(R\) are equal: (1) the Krull-Gabriel dimension of \(R\); (2) the Cantor-Bendixson rank of the Ziegler spectrum over \(R\); (3) the \(m\)-dimension of the lattice of all positive-primitive formulae over \(R\).