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Fun with tensor products


This dissertation addresses various questions in noncommutative ring theory that may be solved with tensor product constructions. First, we consider division rings finite dimensional over their center. Using Kaplansky's Theorem one may show that any subdivision ring will also be finiteover its center. We present an unpublished proof of this theorem, which generalizes to a new result: If D is a division ring algebraic over its center, Z, then any subdivision ring, Dʹ, will also be algebraic over its center, Zʹ. In the following chapter we look at the Martindale ring of quotients of a ring. A ring, A, is called centrally closed if the center of its Martindale ring of quotients is equal to the center of A. If A is centrally closed, then any ideal of A[otimes]k B will contain an ideal of the form I[otimes]k J. We extend this result to show that prime ideals of A[otimes]k B contain ideals of the form P[otimes]k B$ or A[otimes]k Q for P and Q prime ideals of A and B, respectively. Next, we consider Jacobson rings, and we show that a simple ring tensored with a Jacobson ring remains Jacobson. This determines new classes of Jacobson rings, broadening results of Jordan, Goodearl and Warfield. Finally, we look at the classical Krull dimension of polynomial rings over a noncommutative coefficient ring. We show that if R is centrally closed and has classical Krull dimension 1, then R[chi] must have classical Krull dimension 2

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