Math Colloquium: Keith Pardue, NSA

From Ideals to Congruences -
A basic theorem about a ring R is that there is a natural bijection between the ideals of R, and the congruences on R — the equivalence relations that are compatible with addition and multiplication. This theorem breaks when we consider semirings, which are like rings but need not have additive inverses. Semirings have received a lot of attention recently, with attempts to develop an extension of algebraic geometry for commutative rings to also include commutative semirings. Whether one should consider “prime ideals” or “prime congruences” is a significant question, part of which is just what we mean by prime congruence anyways!

I will discuss these questions for the most basic semiring of all, the semiring N of natural numbers with ordinary addition and multiplication. I’ll describe Vandiver’s classification of congruences on N, identify an “extra” prime congruence, and give a multiplication law on the congruences of N for which congruences enjoy unique factorization into prime congruences.