Idempotents in group rings

The von Neumann finiteness problem for k[G] is still open. Kaplansky proved it in characteristic zero. He used the nonvanishing of the trace: tr(e) = 0 implies e = 0 for any idempotent e ∊ k[G]. Assume now that char k = p > 0. Now tr can vanish on nonzero idempotents. Instead, we study the lifted trace ltr. For e = e² ∊ k[G], define ltr(e) by ê(1) where ê = Σ{x ∊ G}ê(x)x lifts e. Here ê is an infinite series with |ê(x)|_p→0, where each ê(x) lives in the Witt vector ring of k. We prove that ltr(e) depends on e only, it is a p-adic integer and ltr(e) = ltr(f) if f is equivalent to e. Also ltr(e) ∊ Q and ltr(e) = 0 implies e = 0 if G is polycyclic-by-finite. We conjecture that -log_p|ltr(e)|_p < |supp(e)|. We prove this for e central and for e = e² ∊ k[G] with |G| ≤ 30. In the last section, we give the example of an idempotent e such ath supp(f) is infinite for all f ~ e. Finally we estimate |<supp(e)>| for central idempotents e.

Name thesis:degree_name

Ph. D.

Level thesis:degree_level

doctoral

Discipline thesis:degree_discipline

Mathematics

Department dc:contributor.department

Mathematics

Grantor dc:publisher

Virginia Polytechnic Institute and State University

Year dc:date.issued

1982

Author dc:creator

• Marciniak, Zbigniew

Chair dc:contributor.committeechair

• Farkas, Daniel R.

Committee members dc:contributor.committeemember

• Green, E.L.
• Dickman, R.F., Jr.
• Feustel, Charles D.
• Hannsgen, Kenneth B.

Statement dc:rights

• In Copyright

Licence dc:rights.uri

• http://rightsstatements.org/vocab/InC/1.0/

Language dc:language.iso

en_US

Handle dc:identifier.uri

http://hdl.handle.net/10919/80271

OAI identifier oai:identifier

oai:vtechworks.lib.vt.edu:10919/80271