## Mathematics Ph.D. Dissertations

#### Title

Clean Rings & Clean Group Rings

2013

Dissertation

#### Degree Name

Doctor of Philosophy (Ph.D.)

Mathematics

#### First Advisor

Warren McGovern, Ph.D.

#### Second Advisor

Rieuwert Blok, Ph.D.

#### Third Advisor

Sheila Roberts, Ph.D. (Committee Member)

#### Fourth Advisor

Mihai Staic, Ph.D. (Committee Member)

#### Abstract

A ring is said to be clean if each element in the ring can be written as the sum of a unit and an idempotent of the ring. More generally, an element in a ring is said to be clean if it can be written as the sum of a unit and an idempotent of the ring. The notion of a clean ring was introduced by Nicholson in his 1977 study of lifting idempotents and exchange rings, and these rings have since been studied by many different authors.

In our study of clean rings, we classify the rings that consist entirely of units, idempotents, and quasiregular elements. It is well known that the units, idempotents, and quasiregular elements of any ring are clean. Therefore any ring that consists entirely of these types of elements is clean. We prove that a ring consists entirely of units, idempotents, and quasiregular elements if and only if it is a boolean ring, a local ring, isomorphic to the direct product of two division rings, isomorphic to the full matrix ring M2(D) for some division ring D, or isomorphic to the ring of a Morita context with zero pairings where both of the underlying rings are division rings. We also classify the rings that consist entirely of units, idempotents, and nilpotent elements.

In our study of clean group rings, we show exactly when the group ring Z(p)Cn is clean, where Z(p) is the localization of the integers at p, and Cn is the cyclic group of order n. It is well known that the group ring Z(7)C3 is not clean even though the group ring Z(p)C3 is quasiclean, semiclean, and Σ-clean for any prime p, and 2-clean for any prime p ≠ 2. We prove that Z(p)C3 is clean if and only if p ≢ 1 modulo 3. More generally, we prove that the group ring Z(p)Cn is clean if and only if p is a primitive root of m, where n = pkm and p does not divide m. We also consider the problems of classifying the groups G whose group rings RG are clean for any clean ring R, and of classifying the rings R such that the group ring RG of any locally finite group G over the ring R is clean.

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