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The dissertation focuses on the structure of π-regular (regular) rings.

The cardinality of the set of units, and of the set of equivalence classes of primes in non-trivial Euclidean domains is discussed with reference to the categories "finite" and "infinite." It is shown that no Euclidean domains exist for which both of these sets are finite. The other three combinations are possible and examples are given. For the more general Euclidean rings, the first combination is possible and examples are likewise given. Prime factorization is also discussed in both Euclidean rings and Euclidean domains. For Euclidean rings, an alternative definition of prime elements in terms of associates is compared and contrasted to the usual definitions.

This thesis investigates some of the properties of power series rings. The material is divided into three chapters. In Chapter I, some of the basic concepts of rings which are a prerequisite to an understanding of the definitions and theorems which follow are stated. Simple properties of power series rings are developed in Chapter II. Many properties of a ring R are preserved when we attach the indeterminant x to form the power series ring R[[x]]. Further results of power series rings are examined in Chapter III. An important result illustrated in this chapter is that power series rings possess some of the properties of rings of polynomials.

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An exploratory investigation of tree growth and climate relationships in Abies religiosa from Michoacan, Mexico, produced the first crossdated and standardized tree-ring chronology from the North American tropics. Pearson correlation coefficients and principal components response function analysis were employed. Results indicate that ring-width series from this species have moderately high signal-to-noise ratio (S/N = 13.42). A substantial percentage of the ring-width signal can be explained by instrumented monthly climate data, particularly spring precipitation and winter temperature. Although correlation between climate data and the tree-ring measurements indicate that growth of Abies religiosa is highly influenced by year-to-year climate variation, longer climate records and tree-ring chronologies are needed from this tropical region to improve understanding of climate -tree growth relationships, and for dendroclimatic reconstruction.