Abstract

Let R be an associative, unital ring. One topic of interest is to assert R has some property, then determine what properties the polynomial ring R[x] and the power series ring R[[x]] are guaranteed to have. There are many relaxations of commutativity in rings. For example, a ring is said to be symmetric if abc = 0 implies acb = 0 for all a, b, c ∈ R. We discuss a certain construction of symmetric rings that we hope will yield the existence of a symmetric ring R such that the set of nilpotent elements in the power series ring R[[x]] does not form an ideal – a much weaker condition than symmetry. We also construct rings that are semicommutative and reversible: R is semicommutative if ab = 0 implies arb = 0 for all a, b, r ∈ R and R is reversible if ab = 0 implies ba = 0 for all a, b ∈ R. Along the way we will touch on reductions systems, Mathematica and Python coding, and SAT solvers.

Degree

College and Department

Computational, Mathematical, and Physical Sciences; Mathematics

Rights

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