#WCWinSTEM: Ranthony A. C. Edmonds, M.S.

Ranthony A. C. Edmonds is a mathematician whose research interests are in a branch of abstract algebra called commutative ring theory!

We’re honored to feature Ranthony A. C. Edmonds as this week’s #WCWinSTEM. Currently a 5th year Ph.D. candidate in the Department of Mathematics at the University of Iowa, Ranthony is excited to tell us more about her journey!

Responses may be edited for clarity and brevity.

Where did you go to school?

• B.S. Mathematics, University of Kentucky, Lexington, KY
• B.A. English with a minor in African American Studies, University of Kentucky, Lexington, KY
• M.S. Mathematical Studies, Eastern Kentucky University, Richmond, KY
• M.S. Mathematics, University of Iowa, Iowa City, IA
• Ph.D. Pure Mathematics (in progress), University of Iowa, Iowa City, IA

What do you do right now?

I am a 5th year PhD candidate at the University of Iowa. My research area is in a branch of abstract algebra that focuses on mathematical objects called rings. A ring is an algebraic structure with two binary operations that generalizes addition and multiplication to both numerical and non-numerical objects. A classic example of a ring is the set of integers. The integers have an additional property, called commutativity, which means that multiplication is the same forwards and backwards. Thus, if we take any two integers, for example 2 and 3, we know that 2*3=3*2. Also, if we take the product of any two integers, call them a and b, if their product a*b=0 we know that a or b must be 0. Any commutative ring that possesses this property is called an integral domain. If there exist two nonzero elements, however, whose product is zero, we call such elements zero divisors. I study factorization in commutative rings with zero divisors.

Factorization theory is concerned with the decomposition of mathematical objects, and its applications are far reaching. Such an object could be a polynomial, a number in the set of integers, or more generally an element in a ring. When we factor an element we reduce it to a product of its basic building blocks, called irreducible elements. For example, at an elementary level, cryptography is the study of factoring very large numbers into the products of building blocks called prime numbers. Authors agree on definitions for the simplest elements of integral domains, such as primes and irreducibles, as well as certain ring theoretic properties like unique factorization that stem from these definitions. Many authors have worked on generalizing this theory to commutative rings with zero divisors, however this theory is much less uniform. The presence of zero divisors has led different authors to establish different definitions of irreducible elements, which in turn leads to different factorization techniques based on what the basic building blocks are considered to be.

Jedidah Isler