Determinantal ideal

Remarks

When $k$ is algebraically closed, the zero set of $I_t(X)$ in $k^{mn}$ is the set of $m \times n$ matrices of rank less than $t$. For the generic matrix, $I_t(X)$ is a prime ideal of height $(m - t + 1)(n - t + 1)$, and $R/I_t(X)$ is Cohen–Macaulay.

Example: for $m = 2$, $n = 3$ and $t = 2$, the ideal $I_2(X)$ is generated by the three minors $x_{11} x_{22} - x_{12} x_{21}$, $x_{11} x_{23} - x_{13} x_{21}$ and $x_{12} x_{23} - x_{13} x_{22}$. Its height is $(2 - 2 + 1)(3 - 2 + 1) = 2$.

Non-example: in the same ring, the principal ideal $(x_{11} x_{22} - x_{12} x_{21})$ is generated by one $2 \times 2$ minor only. It has height $1$ by Krull's principal ideal theorem, so it is not $I_2(X)$, which has height $2$.