Commutative algebra

In this dissertation, we consider six Prufer-like conditions on acommutative ring R. These conditions form a hierarchy. Being a Prufer ring is not a local property: a Prufer ring may not remain a Prufer ring when localized at a prime or maximal ideal. We introduce a seventh condition based on this fact and extend the hierarchy. All the conditions of the hierarchy become equivalent in the case of a domain, namely a Prufer domain. We also seek the relationship of the hierarchy with strong Prufer rings.

This thesis was motivated by the idea to generalize
Spitzer's identity to the non-commutative case. For this
purpose the concept of an "operator type'' is introduced,
which reveals in an elementary way the relationship between
operator identities and functional equations.
Baxter and Reynolds operators are studied using this
approach. Then a non-commutative Spitzer identity is derived
and applied to generalized shift operators. As another
application we give non-commutative analogs of some
formulas of Euler.

For D an integral domain with field of fractions K and E a subset of K, the ring Int (E,D) = {f e K[X]lf (E) C D} of integer-valued polynomials on E has been well studies. In particulare, when E is a finite subset of D, Chapman, Loper, and Smith, as well as Boynton and Klingler, obtained a bound on the number of elements needed to generate a finitely generated ideal of Ing (E, D) in terms of the corresponding bound for D. We obtain analogous results for Int (r) (E, D) - {f e K [X]lf(k) (E) c D for all 0 < k < r} , for finite E and fixed integer r > 1. These results rely on the work of Skolem [23] and Brizolis [7], who found ways to characterize ideals of Int (E, D) from the values of their polynomials at points in D. We obtain similar results for E = D in case D is local, Noetherian, one-dimensional, analytically irreducible, with finite residue field.