Characterizing Zero Divisors of Group Rings
Abstract
The Atiyah Conjecture originates from a paper written 40 years ago by Sir Michael Atiyah,
a famous mathematician and Fields medalist. Since publication of the paper, mathematicians
have been working to solve many questions related to the conjecture, but it is still open.
The conjecture is about certain topological invariants attached to a group G. There are
examples showing that the conjecture does not hold in general. These examples involve
something like the lamplighter group. We are interested in
looking at examples where this is not the case. We are interested in the specific case where
G is a finitely generated group in which the Pr'ufer group can be embedded as the center.
The Pr'ufer group is a p-group for some prime p and its finite subgroups have unbounded
order, in particular the finite subgroups of G will have unbounded order.
To understand whether any form of the Atiyah conjecture is true for G, it will first help to
determine whether the group ring kG of the group G has a classical ring of quotients for
some field k. To determine this we will need to know the zero divisors for the group ring
kG. Our investigations will be divided into two cases, namely when the characteristic of the
field k is the same as the prime p for the Pr'ufer group and when it is different.
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- Masters Theses [18655]