Date of Final Oral Examination (Defense)

5-2011

Type of Culminating Activity

Thesis

Degree Title

Master of Science in Mathematics

Department

Mathematics

Major Advisor

Jens Harlander, Ph.D.

Abstract

In 1964, Paul Cohn showed that if F is a finitely-generated free group, and Q a field, then all ideals in the group ring Q[F] are free as Q[F]-modules. In particular, all finitely-generated submodules of free Q[F]-modules are free. In 1990, Cynthia Hog-Angeloni reproved this theorem using techniques from geometric group theory. Leaning on Hog-Angeloni's methods, we prove an analogous statement for crossed products D * F, with D a division ring.

With this result in hand, we prove that if G = H⋊F, the semi-direct product of H with F, so that the group ring D[G] may be localized at the sub-group ring k[H]―{0), then the resulting localized group ring also has the property that finitely-generated submodules of free modules are free.

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