In ring theory, a branch of mathematics, the conductor is a measurement of how far apart a commutative ring and an extension ring are. Most often, the larger ring is a domain integrally closed in its field of fractions, and then the conductor measures the failure of the smaller ring to be integrally closed. The conductor is of great importance in the study of non-maximal orders in the ring of integers of an algebraic number field. One interpretation of the conductor is that it measures the failure of unique factorization into prime ideals.