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

By Paul C. Eklof

This is often a longer remedy of the set-theoretic innovations that have reworked the research of abelian staff and module thought over the past 15 years. a part of the ebook is new paintings which doesn't look in other places in any shape. furthermore, a wide physique of fabric which has seemed formerly (in scattered and occasionally inaccessible magazine articles) has been commonly transformed and in lots of circumstances given new and more advantageous proofs. The set idea required is thoroughly constructed with algebraists in brain, and the independence effects are derived from explicitly acknowledged axioms. The e-book includes workouts 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 common sense.

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8, M* is free. (Y) = --f M* as follows: a ifaEY 0 otherwise and let ‘p(a)= GD. Now a E Y for some Y E I , so G ( 2 ) = a for all 2 E U y . Therefore it is clear that ‘p is an embedding. , it follows that M is free. 11 Corollary. If. is a n LwlW-compactcardinal, then every K-free abelian group (of arbitrary cardinality) is free. 3. $4. Cubs a n d s t a t i o n a r y sets Our theme in this section will be the description of “relatively large” subsets of a limit ordinal y; we will have analogs of “sets of measure 1” (the closed unbounded sets, or cubs) and of “sets of non-zero measure” (the stationary sets).

The notation we are using is ambiguous in that the operations of cardinal arithmetic and ordinal arithmetic are denoted by the same symbols. But it is usually clear from the context which operation is intended. + 52. Filters and large cardinals Throughout this section I will denote an arbitrary (infinite) set. 1 Definition. A filter on I is a subset D of P ( I ) satisfying: (1) 8 $! D ,I E D ; ( 2 ) if X , Y E D,then X n Y E D; (3) if X E D and X 2 Y C I , then Y E D. If Y is a subset of I , the principal filter generated b y Y is the filter D E P ( I ) :Y C_ X I .

Z ( n E w \ (0)). u, where a, E Z. Indeed, given x = (. . ,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).