The Existence of Large E(R)-Algebras That Are Sharply Transitive Modules

Abstract

Let R be any principal ideal domain of cardinality < 2^ℵ₀ , but not a field. We will construct arbitrarily large E(R)-algebras A (of size ≥ 2^ℵ₀ ) which are at the same time principal ideal domains over R. It follows that the automorphism group of the R-module _R A acts sharply transitive on the pure elements of _R A. In answering a question of Emmanuel Dror Farjoun, the existence of such large uniquely transitive (UT-modules) for R = ℤ was shown in Göbel and Shelah (2004). The new method, passing first through ring theory, simplifies the arguments; this idea, using localizations, comes from Herden (2005). We applied it recently in Göbel and Herden (2007b) to find such E(R)-algebras of size ≤ 2^ℵ₀ .

Key Words:

• Automorphism groups
• E-rings
• Homotopy theory
• Indecomposable modules

2000 Mathematics Subject Classification:

• Primary 20K20, 20K30
• Secondary 16S60, 16W20

ACKNOWLEDGMENT

This work is supported by the project No. I-706-54.6/2001 of the German–Israeli Foundation for Scientific Research & Development.

Notes

Communicated by K. M. Rangaswamy