$$\aleph_n$$ -Free Modules with Trivial Duals

Abstract

In the first part of this paper we introduce a simplified version of a new Black Box from Shelah [11] which can be used to construct complicated ℵn-free abelian groups for any natural number n ∈ N. In the second part we apply this prediction principle to derive for many commutative rings R the existence of ℵn-free R-modules M with trivial dual M * = 0, where M * = Hom(M, R). The minimal size of the ℵn-free abelian groups constructed below is n, and this lower bound is also necessary as can be seen immediately if we apply GCH. Mathematics Subject Classification (2000). Primary 13C05, 13C10, 13C13, 20K20, 20K25, 20K30; Secondary 03E05, 03E35.

ℵn-FREE MODULES OVER COMPLETE DISCRETE VALUATION DOMAINS WITH ALMOST TRIVIAL DUAL

“…. , 0η k * for some η ∈ (not just 0η k * as in [9]). The changes of the proof of the next theorem are minor and thus follow easily from the proof of [9, Theorem 2.4].…”

“…From the set-theoretic version of the Black Box 2.5 follows as in [9,Theorem 3.3] (by easy modification) its algebraic counterpart, which we want to apply in Section 3.…”

“…The ℵ n -free black box from [9]. Here we outline the new combinatorial principle from [13], which was simplified in [9]. It is designed to construct ℵ n -free abelian groups in ZFC (without any additional axioms).…”

Separable $\aleph_k$-free modules with almost trivial dual

“…Only in 2007 Shelah [15] introduced a new, more powerful version of his Black Box principle, which allowed him to construct ℵ n -free abelian groups with trivial dual. This breakthrough immediately led to other constructions of ℵ n -free groups and modules with different algebraic properties like almost trivial dual or prescribed endomorphism ring (see for example [6], [7] and [8]).…”

“…In [7], the authors construct a class of ℵ k -free R-modules M for a fixed natural number k > 1, where R is a countable domain, but not a field. Moreover, these modules have trivial dual , i.e.…”