By Andrei V. Kelarev, R. Gobel, K. M. Rangaswamy, P. Schultz, C. Vinsonhaler

This quantity provides the complaints from the convention on Abelian teams, jewelry, and Modules (AGRAM) held on the collage of Western Australia (Perth). integrated are articles in keeping with talks given on the convention, in addition to a couple of specifically invited papers. The court cases are devoted to Professor Laszlo Fuchs. The publication encompasses a tribute and a evaluate of his paintings by means of his long-time collaborator, Professor Luigi Salce. 4 surveys from top specialists stick to Professor Salce's article.They current fresh effects from energetic examine parts: errors correcting codes as beliefs in workforce earrings, duality in module different types, automorphism teams of abelian teams, and generalizations of isomorphism in torsion-free abelian teams. as well as those surveys, the quantity comprises 22 learn articles in diversified parts hooked up with the subjects of the convention. The parts mentioned comprise abelian teams and their endomorphism earrings, modules over a variety of earrings, commutative and non-commutative ring conception, kinds of teams, and topological features of algebra. The ebook deals a finished resource for fresh study during this energetic zone of research

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**Extra info for Abelian Groups, Rings and Modules: Agram 2000 Conference July 9-15, 2000, Perth, Western Australia**

**Example text**

It remains to analyze the case a + b = 2. Since N is nonabelian it follows that a = b = 1. However, since N is nilpotent, 7l is a central subgroup in N, which implies that N is abelian. Contradiction. D 2 Thurston Norm In this chapter, we prove the following theorem. 1 (W. Thurston [Thu85]). Suppose that M is a compact atoroidal orientable 3-manifold such that rank H2(M, oM; Z) 2:: 2. Then M contains an embedded superincompressible surface that is not a fiber in a fibration of M over § 1 and that represents a nontrivial element of H2(M, oM; Z).

Call the resulting manifold M. 10 but F c M is not incompressible (because (1) fails) . Show that the manifold M contains no incompressible surfaces. To prove Thurston's Hyperbolization Theorem, we shall need a somewhat more restrictive definition than that of an incompressible surface. A compressing annulus for (F, a F) c (M, aM) is an embedded annulus such that (i) i*: 7rt(A) ~ Jrt(M) is an embedding; (ii) AnF={O}x§ 1 =y; (iii) y is not isotopic to aF in F. 12. Take M compressing annulus. 5}.

Suppose that a, b are elements of a group G. 2) Note that if p = q, then the relation reduces to the "torus" relation cb=bc, where c :=aP. If p = -q, then we get the "Klein bottle" relation cb=bc- 1 , wherec:=aP. If IPI =f. lql and a =f. 2) is nonelementary. The group BSp,q given by the presentation is called a Baumslag-Solitar group. 53. Show that B Sz 3 contains Z 2. 10. 54 (W. Jaco, P. Shalen [JS79]). Suppose that G is the fundamental group of a Haken 5 3-manifold M . Then there are no nontrivial elements a, b E G that satisfy a nonelementary Baumslag-Solitar relation.