Georgia Institute of Technology College of Sciences

Let I be an ideal in a polynomial ring R := F[x_n,...,x_1] Let A := R/I be the corresponding quotient ring, and let Q(A) be its field of fractions. The integral closure C(A, Q(A)) of A in Q(A) is a subring of the latter. But it is often given as a separate quotient ring, a presentation.Surprisingly, different computer algebra systems (Magma, Macaulay2, and Singular) choose to produce very different presentations. Some of these opt for presentations that have seductive forms, but miss the most important, namely a form that allows for determining when elements of Q(A) are in C(A,Q(A)). This is called membership and is directly related to determining isomorphism.