"Pure mathematics is, in its way, the poetry of logical ideas."
-Albert Einstein
Doctoral Work
My research focus is in a branch of abstract algebra called commutative ring theory. In particular, I am interested in the factorization properties of commutative rings, especially those with zero-divisors. For integral domains, factorization theory has been well established. There are standard definitions for the building blocks of such rings, such as irreducible and prime elements, as well as certain ring theoretic properties, like unique factorization. When discussing factorization in commutative rings with zero divisors however, things become more complicated. For example, the presence of zero-divisors has led to different definitions of irreducible and associate elements by several authors, and different factorization techniques have led to several types of unique factorization rings.
My research focuses on how factorization properties behave with respect to certain extensions. Of particular interest is the extension of a commutative ring R with identity to the polynomial ring extension R[X]: if one of these rings has a certain factorization property, does the other? If the answer is no, we would like to have nice counterexamples, and polynomial rings often give much simpler examples than you might see for arbitrary commutative rings.
My dissertation has the goal of creating an overarching theory for factorization in polynomial rings with zero divisors. I aim to do this by focusing on their behavior with respect to many important factorization properties such as unique factorization, bounded factorization, finite factorization, the ascending chain condition on principal ideals (ACCP), and atomicity. To this end, I have focused primarily on unique factorization.
My research focuses on how factorization properties behave with respect to certain extensions. Of particular interest is the extension of a commutative ring R with identity to the polynomial ring extension R[X]: if one of these rings has a certain factorization property, does the other? If the answer is no, we would like to have nice counterexamples, and polynomial rings often give much simpler examples than you might see for arbitrary commutative rings.
My dissertation has the goal of creating an overarching theory for factorization in polynomial rings with zero divisors. I aim to do this by focusing on their behavior with respect to many important factorization properties such as unique factorization, bounded factorization, finite factorization, the ascending chain condition on principal ideals (ACCP), and atomicity. To this end, I have focused primarily on unique factorization.
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