By Paul C. Eklof

This can be a longer therapy of the set-theoretic innovations that have remodeled the learn of abelian staff and module idea during the last 15 years. a part of the booklet is new paintings which doesn't seem somewhere else in any shape. furthermore, a wide physique of fabric which has seemed formerly (in scattered and infrequently inaccessible magazine articles) has been largely remodeled and in lots of situations given new and more suitable proofs. The set idea required is thoroughly constructed with algebraists in brain, and the independence effects are derived from explicitly said axioms. The publication comprises workouts and a consultant to the literature and is appropriate 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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**Almost Free Modules: Set-Theoretic Methods**

This is often a longer therapy of the set-theoretic suggestions that have remodeled the learn of abelian team and module idea over the past 15 years. a part of the publication 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 generally remodeled and in lots of circumstances given new and more advantageous proofs.

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

**Example text**

Suppose F is a K-complete filter on which contains the cofinite filter. Let S K such that K K = X+ \ S $! F . 11. SET THEORY 38 Then there is a decomposition of S , S = UpElcSp,into n pairwise disjoint subsets Sp such that for all ,f3, K \ Sp 6 F. For each a < 6, choose a surjection g,: X + a. (Here we use the Axiom of Choice,) For all v E A, ,f3 E n, let s; = { a E S:g,(v) = p}. We claim that there is a v such that { S i : ,d E n, n \ S i $ F } has cardinality n. fix p and let Yp = u{Si: v E A}; then Yp = { a E ~ : >ap } n S , since the g,’s are surjective.

If 6 is weakly compact and M is a 6-free abelian group of cardinality 6 ,then A4 is free. 10, choose an appropriate 6-complete subalgebra L? of I and form an "ultraproduct" with respect to an ultrafilter D on L?. 1 niEI 2. If D is a principal ultrafilter, generated by { j } , then M;/D is canonically isomorphic to Mj via the map which takes U D to u ( j ) . 3. Given a finitely additive measure: p: P ( I ) + (0, l} there is an ultrafilter D such that p = p ~ . 46 11. SET THEORY 4. D is called a regular ultrafilter on I if there exists a subset X of D of cardinality 111 such that each element of I belongs to only finitely many members of X .

Y E x f X implies y E X . Given any set S , define by induction So = S and S,+l =US,. Then U{S,:n E w } is transitive and is the smallest transitive set containing S; it is called the transitive closure of S , and is denoted TC(S). The class, Ord, of ordinal numbers (or ordinals) is defined to be the class of all sets which are transitive and well-ordered by E. It may be proved that every member of an ordinal is an ordinal (that is, Ord is a transitive class) and for any two different ordinals a and p, either a E ,B or p E a.