TY - JOUR
T1 - Decompositions of reflexive modules
AU - Göbel, Rüdiger
AU - Shelah, Saharon
PY - 2001/3/1
Y1 - 2001/3/1
N2 - In a recent paper [11] we answered to the negative a question raised in the book by Eklof and Mekler [8, p. 455, Problem 12] under the set theoretical hypothesis of ◇א1 which holds in many models of set theory. The Problem 12 in [8] reads as follows: If A is a dual (abelian) group of infinite rank, is A ≅ A ⊕ ℤ? The set theoretic hypothesis we made is the axiom ◇א1 which holds in particular in Gödel's universe as shown by R. Jensen, see [8]. Here we want to prove a stronger result under the special continuum hypothesis (CH). The question in [8] relates to dual abelian groups. We want to find a particular example of a dual group, which will provide a negative answer to the question in [8]. In order to derive a stronger and also more general result we will concentrate on reflexive modules over countable principal ideal domains R. Following H. Bass [3] an R-module G is reflexive if the evaluation map σ : G → G** is an isomorphism. Here G* = Hom (G, R) denotes the dual group of G. Guided by classical results the question about the existence of a reflexive R-module G of infinite rank with G ≇ G ⊕ R is natural, see [8, p. 455]. We will use a theory of bilinear forms on free R-modules which strengthens our algebraic results in [11]. Moreover we want to apply a model theoretic combinatorial theorem from [14] which allows us to avoid the weak diamond principle. This has the great advantage that the used prediction principle is still similar to ◇א1 but holds under CH. This will simplify algebraic argument for G ≇ G ⊕ R.
AB - In a recent paper [11] we answered to the negative a question raised in the book by Eklof and Mekler [8, p. 455, Problem 12] under the set theoretical hypothesis of ◇א1 which holds in many models of set theory. The Problem 12 in [8] reads as follows: If A is a dual (abelian) group of infinite rank, is A ≅ A ⊕ ℤ? The set theoretic hypothesis we made is the axiom ◇א1 which holds in particular in Gödel's universe as shown by R. Jensen, see [8]. Here we want to prove a stronger result under the special continuum hypothesis (CH). The question in [8] relates to dual abelian groups. We want to find a particular example of a dual group, which will provide a negative answer to the question in [8]. In order to derive a stronger and also more general result we will concentrate on reflexive modules over countable principal ideal domains R. Following H. Bass [3] an R-module G is reflexive if the evaluation map σ : G → G** is an isomorphism. Here G* = Hom (G, R) denotes the dual group of G. Guided by classical results the question about the existence of a reflexive R-module G of infinite rank with G ≇ G ⊕ R is natural, see [8, p. 455]. We will use a theory of bilinear forms on free R-modules which strengthens our algebraic results in [11]. Moreover we want to apply a model theoretic combinatorial theorem from [14] which allows us to avoid the weak diamond principle. This has the great advantage that the used prediction principle is still similar to ◇א1 but holds under CH. This will simplify algebraic argument for G ≇ G ⊕ R.
UR - http://www.scopus.com/inward/record.url?scp=0035530999&partnerID=8YFLogxK
U2 - 10.1007/s000130050557
DO - 10.1007/s000130050557
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AN - SCOPUS:0035530999
SN - 0003-889X
VL - 76
SP - 166
EP - 181
JO - Archiv der Mathematik
JF - Archiv der Mathematik
IS - 3
ER -