A Krull–Schmidt Theorem for Noetherian Modules

✍ Scribed by Gary Brookfield

• Publisher: Elsevier Science
• Year: 2002
• Tongue: English
• Weight: 102 KB
• Volume: 251
• Category: Article
• ISSN: 0021-8693
• DOI: 10.1006/jabr.2002.9142

✦ Synopsis

We prove a version of the Krull-Schmidt theorem which applies to Noetherian modules. As a corollary we get the following cancellation rule:

then there are modules A ≤ A and B ≤ B such that A ∼ = B and len A = len A = len B = len B. Here the ordinal valued length, len A, of a module A is as defined in G. Brookfield [Comm. Algebra 30 (2002), 3177-3204] and T. H. Gulliksen [J. Pure Appl. Algebra 3 (1973), 159-170]. In particular, A B A , and B have the same Krull dimension, and A/A and B/B have strictly smaller Krull dimensions than A and B.  2002 Elsevier Science (USA) 1. [4] If C is Artinian, then A ∼ = B.