By Paul C. Eklof

ISBN-10: 0444885021

ISBN-13: 9780444885029

This can be a longer remedy of the set-theoretic innovations that have reworked the examine of abelian crew and module idea during the last 15 years. a part of the ebook is new paintings which doesn't look somewhere else in any shape. furthermore, a wide physique of fabric which has seemed formerly (in scattered and infrequently inaccessible magazine articles) has been commonly transformed and in lots of situations given new and better proofs. The set conception required is punctiliously built with algebraists in brain, and the independence effects are derived from explicitly said axioms. The publication comprises routines and a consultant to the literature and is acceptable to be used in graduate classes or seminars, in addition to being of curiosity to researchers in algebra and good judgment.

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**Extra resources for Almost Free Modules: Set-Theoretic Methods**

**Example text**

The converse is not true, as we shall see, but the corollary to the following theorem says that the existence questions are equivalent. 11 Theorem. Assume that there is an w-measurable cardinal and let K be the first one. Then 26 11. SET THEORY (i) every cardinal 2 K is w-measurable; and (ii) for every set I, every wl-complete ultrafilter on I is K-complete. (i) If D is a non-principal wl-complete ultrafilter on K and X is a cardinal 2 K , we can define a non-principal q-complete ultrafilter D’ on A, as follows.

We claim that if t E T is such that {s E T : t < s} is infinite, then there is tl > t so that {s f T : t l < s} is infinite. Given this claim, we can inductively choose an increasing sequence t o < tl < t 2 . . so that for all n, {s E T : t , < s} is infinite: we can take t o to be the minimum element of the tree and the claim allows us to pick tn+l if t , has been chosen. Then {t,: n E w } is contained in a branch. As for the claim, suppose t is as in the claim. There is an m E w such that t has exactly rn immediate successors { s k : k < m } .

X, nZ, . ) E 2, we define a , inductively so that for all rn,x,! n ! Z. ) (ii) Let M be the Z-adic closure of Z(") in Z". Suppose H is a reduced torsion-free group. re given the Z-adic topology. Thus Hom(M, H ) is naturally isomorphic to a subgroup of Hom(Z("), H ) % H". ) -+ i. If f(e,) # 0, for infinitely many a , then f [ M ] Z. So Hom(A4, Z) is naturally isomorphic to Z("). CHAPTER I1 SET THEORY In this chapter we work within ordinary set theory, ZFC (defined in section 1). Although we presume some familiarity with ordinal and cardinal numbers and other elementary concepts from axiomatic set theory, we briefly review these notions, without proofs, in the first section.

### Almost Free Modules: Set-Theoretic Methods by Paul C. Eklof

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