The Krull–Gabriel Dimension of a Serial Ring

Abstract

We prove that every serial ring R has the isolation property: every isolated point in any theory of modules over R is isolated by a minimal pair. Using this we calculate the Krull–Gabriel dimension of the module category over serial rings. For instance, we show that this dimension cannot be equal to 1.

Key Words:

• Serial ring
• Krull–Gabriel dimension
• Isolation property

1991 Mathematics Subject Classification:

• 16L30
• 16B70

Acknowledgment

This paper was written during the visit of the author to the University of Manchester supported by EPSRC grant GR/R44942/01. He would like to thank the University for the kind hospitality.