Direct Sum Decompositions of Torsion-Free Finite Rank Groups by Theodore G. Faticoni

By Theodore G. Faticoni

With lots of new fabric no longer present in different books, Direct Sum Decompositions of Torsion-Free Finite Rank teams explores complicated subject matters in direct sum decompositions of abelian teams and their results. The booklet illustrates a brand new manner of learning those teams whereas nonetheless honoring the wealthy historical past of particular direct sum decompositions of groups.Offering a unified method of theoretic thoughts, this reference covers isomorphism, endomorphism, refinement, the Baer splitting estate, Gabriel filters, and endomorphism modules. It exhibits easy methods to successfully learn a bunch G by means of contemplating finitely generated projective correct End(G)-modules, the left End(G)-module G, and the hoop E(G) = End(G)/N(End(G)). for example, one of many evidently taking place houses thought of is whilst E(G) is a commutative ring. sleek algebraic quantity conception offers effects about the isomorphism of in the neighborhood isomorphic rtffr teams, finitely trustworthy S-groups which are J-groups, and every rtffr L-group that could be a J-group. The ebook concludes with beneficial appendices that include heritage fabric and diverse examples.

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Example text

The next lemma gives us the relationship between local isomorphism of rtffr groups and the local isomorphism of finitely generated projective modules. Its proof rests on the fact that an additive functor takes multiplication by an integer to multiplication by an integer. It is left as an exercise. 1 Let G be an rtffr group, let H, K ∈ Po (G) be rtffr groups, and let E = End(G). The following are equivalent. 1. H and K are locally isomorphic as groups. 36 CHAPTER 2. MOTIVATION BY EXAMPLE 2. HG (H) and HG (K) are locally isomorphic as right E-modules.

That is, II ∗ = S where I ∗ = {q ∈ QS qI ⊂ S}. 3. Each nonzero ideal of S is a unique product of maximal ideals in S. 4. S is a hereditary Noetherian integral domain. 5. The localization SM is a discrete valuation domain for each maximal ideal M in S. That is, there is an element π ∈ SM such that each ideal of SM has the form π k SM for some integer k > 0. For example, Z is a Dedekind domain as is any pid. The ring of algebraic integers in an algebraic number field is a Dedekind domain. The rtffr ring E is integrally closed if whenever E ⊂ E ⊂ QE is a ring such that E /E is finite then E = E .

4. 5) iff QU ∼ = QV as right QE-modules. ) Semi-prime rtffr rings are closely connected to integrally closed rings as the following result shows. The next two results follow from [10, page 127]. 2 Let E be a semi-prime rtffr ring. There is a finite set of primes {p1 , . . , ps } ⊂ Z such that Ep is a classical maximal order for all p ∈ {p1 , . . , ps }. 3 Suppose that E is a semi-prime rtffr ring. 7 Semi-Perfect Rings The ring E is semi-perfect if 1. E/J (E) is semi-simple Artinian and 2. Given an e¯2 = e¯ ∈ E/J (E) there is an e2 = e ∈ E such that e¯ = e + J (E).

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