Determinantal ideal theorem

History
The result was proved for polynomial rings by Macaulay and for arbitrary Noetherian rings by Eagon.

Statement
Let $$M$$ be a $$p \times q$$ matrix with entries over a Noetherian ring $$R$$. Denote by $$I_k(M)$$ the ideal generated by the $$k \times k$$ minors of $$M$$. Then, the codimension of any prime ideal minimal over $$I_k(M)$$ is at most $$(p - k + 1)(q - k + 1)$$.

The case $$k = p = 1$$ yields Krull's height theorem and the case $$p = q = k = 1$$ yields Krull's principal ideal theorem.