Doctor of Philosophy (PhD), Ohio University, 2011, Mathematics (Arts and Sciences)

We study various connections between the notions of invertibility of elements and linear independence of subsets of algebras over (not necessarily commutative) rings. The main emphasis is on two such notions: invertible and fluid algebras. We introduce a hierarchy of notions about algebras having a basis B consisting entirely of units. Such a basis is called an invertible basis and algebras that have invertible bases are said to be invertible algebras. The conditions considered in that hierarchy include the requirement that for an invertible basis B, the set of inverses B-1 be itself a basis, the notion that B be closed under inverses and the idea that B be closed under products under a slight commutativity requirement. Among other results, it is shown that this last property is unique of group rings. Many examples are considered and it is determined that the hierarchy is for the most part strict. For any field F not equal to F2, all semisimple F-algebras are invertible. Semisimple invertible F2-algebras are fully characterized. Likewise, the question of which single-variable polynomials over a field yield invertible quotient rings of the F-algebra F[x] is completely answered. Connections between invertible algebras and S-rings (rings generated by units) are also explored. While group rings are the archetype of invertible algebras, this notion is general enough to include many other families of algebras. For example, field extensions and all crossed products (including in particular skew and twisted group rings) are invertible algebras. We consider invertible bases B such that for any two elements from B, a scalar multiple of their product belongs to B. Alternatively, one may consider invertible bases with the requirement that for every basis element, a scalar multiple of its inverse must also be in the basis. We refer to these algebras, respectively, as being scalarly closed under products and scalarly closed under inverses. We explore connections between these ide (open full item for complete abstract)

Committee: Sergio Lopez-Permouth (Advisor); Dinh Huynh (Committee Member); Franco Guerrerio (Committee Member); Jeffery Dill (Committee Member) Subjects: Mathematics