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.

Let A and B be commutative rings, and assume A ⊆ B. The conductor of A in B is the ideal

Here B /A is viewed as a quotient of A-modules, and Ann denotes the annihilator. More concretely, the conductor is the set

Because the conductor is defined as an annihilator, it is an ideal of A.

If B is an integral domain, then the conductor may be rewritten as

where ${\displaystyle \textstyle {\frac {1}{a}}A}$ is considered as a subset of the fraction field of B. That is, if a is non-zero and in the conductor, then every element of B may be written as a fraction whose numerator is in A and whose denominator is a. Therefore the non-zero elements of the conductor are those that suffice as common denominators when writing elements of B as quotients of elements of A.

Suppose R is a ring containing B. For example, R might equal B, or B might be a domain and R its field of fractions. Then, because 1 ∈ B, the conductor is also equal to