Determinantal ring

History
The definition that we are using follows Commutative algebra with a view towards algebraic geometry by David Eisenbud.

Definition
A commutative unital ring $$R$$ is said to be a determinantal ring over the commutative unital ring $$S$$ if it can be written as $$S/I$$ where $$I$$ is the ideal generated by the $$r \times r$$ minors of a $$p \times q$$ matrix $$M$$, for some $$p, q, r$$, such that that codimension of $$I$$ in $$S$$ is exactly $$(p-r+1)(q-r+1)$$.

Facts

 * A determinantal ring of type $$(1,1)$$ over a regular ring is termed a complete intersection
 * Any determinantal ring over a Cohen-Macaulay ring is Cohen-Macaulay