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The Rings of Commutative Algebra

A visit with Ian Aberbach, Professor of Mathematics

By LuAnne Roth
Published: - Topics: mathematics

Ian Aberbach confesses that he thought about pursuing English as a major in college, but found the problems in a modern algebra class so engaging that he was drawn inescapably to mathematics instead. Taking that fork in the road has led Aberbach to a career in commutative algebra. During our interview, the math professor patiently allowed me to test the claim that “no question is a stupid question.” When asked to explain his research to the general public, Aberbach admitted that he wasn’t sure whether that was possible, referring to the highly specialized language, concepts, and theory in which his work is situated—concepts that are crucial for algebraists, but challenging for those outside of that subfield to wrap their minds around. In spite of the highly technical language, Aberbach does his best to explain his research in layperson’s terms.

Within the area of algebra, a broad subfield of mathematics, Aberbach studies algebraic objects called rings, especially looking at those rings that are “commutative.” A ring is a set, along with both addition and multiplication operations, which satisfy certain axioms. “Commutative” in this context means, for example, that 3 x 5 is the same as 5 x 3. As a motivating example, think for instance of the integers. Another motivating example is the ring of polynomials, since one can both add and multiply them together. Much of commutative algebra consists of trying to understand the solution set of a collection of polynomials. This set is geometric in nature, but there is an associated ring that can be studied, and these two relate to each other. As a simple example, the solution set of x2 + y2 = 1 is a circle, and there is a related ring which can be used to study this object. As it turns out, the geometric object and the ring somehow match up, so that properties of the ring reveal something about the geometric object and vice versa.

Aberbach’s research poses questions about what properties certain rings have—particularly a kind of ring that has prime characteristics. “Numbers that are prime are very special,” Aberbach explains. “They have a lot of properties that non-prime numbers do not.” In any ring, something in the ring always acts like the number 1 does (that is, 1 x A = A). In some rings unexpected things occur. To illustrate, Aberbach poses the question: “If it’s 11:00 right now, and I want to meet you in 2 hours, then what time do I meet you?” “1:00,” I respond, somehow suspecting that this is a trick question. Why not 13:00? Along this line, there are rings in which sometimes you can add 1 + 1 + 1 and get zero (instead of 3) because they wrap around again, sort of like a clock. When, in a ring, one can add 1 to itself a prime number of times and get zero (e. g., 2, 3, 5), then there are some powerful techniques for studying the ring that are not available otherwise.

When asked whether there were practical applications to his work, Aberbach replied modestly, “Nothing I’ve ever done has any application in the ‘real world’—not applications in the sense that most people would understand them, but answering questions posed by other mathematicians.” Yet the kind of theories Aberbach tackles are taken up by folks in computer science and engineering—where a sound theoretical framework may allow them to quickly solve equations. In that way, the practical applications are several steps removed from the deeply technical and theoretical work in which Aberbach is engaged. If you are able to follow this algebraic logic and want more, I recommend you listen to Aberbach himself as he talks about rings, commutative algebra, and MU’s leading place in this international conversation.