Generalized Analytic Independence

If <!-- MATH ${\mathbf{a}}$ --> is a proper ideal of a commutative ring with unity , a set of elements <!-- MATH ${a_1}, \ldots ,{a_n} \in A$ --> is called <!-- MATH ${\mathbf{a}}$ --> -independent if every form in <!-- MATH $A[{X_1}, \ldots ,{X_n}]$ --> vanishing at <!-- MATH ${a_1}, \ldots ,{a_n}$ --> has all its coefficients in <!-- MATH ${\mathbf{a}}$ --> . <!-- MATH $\sup {\mathbf{a}}$ --> is defined as the maximum number of <!-- MATH ${\mathbf{a}}$ --> -independent elements in <!-- MATH ${\mathbf{a}}$ --> . It is shown that grade <!-- MATH ${\mathbf{a}} \leq \sup {\mathbf{a}} \leq {\text{height }}{\mathbf{a}}$ --> . Examples are given to show that <!-- MATH $\sup {\mathbf{a}}$ --> need take neither of the limiting values and strong evidence is given for the conjecture that it can assume any intermediate value. Cohen-Macaulay rings are characterized by the equality of sup and grade for all ideals (or just all prime ideals). It is proven that equality of sup and height for all powers of prime ideals implies that the ring is (the Serre condition). Finally, independence is related to the structure of certain Rees algebras.