Absolute E-modules

Abstract

Communicated by C.A. Weibel MSC: Primary: 13C05; 13C10; 13C13; 20K20; 20K25; 20K30 Secondary: 03E05; 03E35 a b s t r a c t Let R be a ring with 1 and M a right R-module. Then M is called E-module if Hom Z (R, M) = Hom R (R, M). Thus all homomorphisms between the abelian groups R Z and M Z turn out to be R-homogeneous. If M = R, then R is called an E-ring. It is clear from the definition, that the existence of E-modules requires that R be an E-ring and simple examples of E-rings are all subrings of Q. These structures are studied for more than thirty years in relation to questions concerning properties of modules, non-commutative groups and in the context of algebraic topology, see Casacuberta et al. (2010) [3] and Göbel and Trlifaj (2006) [24]. Constructions of large classes of E-rings were given in Faticoni (1987) [13] and Dugas et al. (1987) [9]; in a subsequent paper Dugas (1991) [6] showed the existence of large E-modules over R. More recently (initiated by Eklof and Shelah (1999) [11]) absolute algebraic structure came into the focus of studies see also Eklof and Mekler (2002) [10, p. 487 ff.]. Their advantage is, that they are robust under changes of the universe as explained in the introduction. We want to show the existence of absolute E-modules. On one hand, from earlier work on absolute E-rings Herden and Shelah (2009) [25] it follows that they must be smaller than κ(ω), which is the first ω-Erdős cardinal. Like measurable cardinals, κ(ω) is a large cardinal and may not exist. On the other hand, for each infinite cardinal λ an absolute E-ring of cardinality λ < κ(ω) exists, as shown in Göbel et al. (2010) [22]. Given such an absolute E-ring R, then we want to construct an absolute Emodule M over R for any cardinality |R| ≤ |M| < κ(ω) and study the decompositions of M.