ABSTRACT
Saharon Shelah Institute of Mathematics, Hebrew University, Jerusalem, Israel and Department of Mathematics, Rutgers University, New Brunswick, NJ, USA shelah@math.huji.ac.il
14.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 14.2 Proof of the Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158
Abstract The aim of this paper is to answer a problem raised in a recent monograph by Robert Colby and Kent Fuller [3, pp. 129, 130] concerning R-torsionless linearly compact R-modules; see the introduction for a precise definition of this class of modules. Over a ring R these modules are particular submodules of products Rκ . Are Z(ω) and P = Zω Z-torsionless linearly compact (for R = Z)? Is this class closed under direct sums? Both questions can be answered to the negative. In fact we show much more and characterize Z-torsionless linearly compact groups: They are the free groups of finite rank. The same result holds for all principal ideal domains which are neither fields nor complete discrete valuation rings. This work is supported by the project No. I-706-54.6/2001 of the German-Israeli Foundation for Scientific Research & Development. Shelah’s list of publications GBSh 834. Subject classifications: 20K20, 20K25, 20K30, 16D90, 16D70.
