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.
The conductor[1] of A in B is the ideal Here B /A is viewed as a quotient of A-modules, and Ann denotes the annihilator.
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.
This applies in particular when B is the ring of integers in an algebraic number field and A is an order (a subring for which B /A is finite).
Then with equality in the case that B is a finitely generated A-module.
Some of the most important applications of the conductor arise when B is a Dedekind domain and B /A is finite.
For example, B can be the ring of integers of a number field and A a non-maximal order.
Or, B can be the affine coordinate ring of a smooth projective curve over a finite field and A the affine coordinate ring of a singular model.
To see some of the difficulties that may arise, assume that J is a non-zero ideal of both A and B (in particular, it is contained in, hence not coprime to, the conductor).
Because B is a Dedekind domain, J is invertible in B, and therefore since we may multiply both sides of the equation xJ ⊆ J by J −1.
But the left-hand side of the above equation makes no reference to A or B, only to their shared fraction field, and therefore A = B.
By extending 1 ∈ OK to a Z-basis, we see that every order O in K has the form Z + cOK for some positive integer c. The conductor of this order equals the ideal cOK.
This identification fails for higher degree number fields.