Lemma 15.8.1. Let $R$ be a ring. Let $A$ be an $n \times m$ matrix with coefficients in $R$. Let $I_ r(A)$ be the ideal generated by the $r \times r$-minors of $A$ with the convention that $I_0(A) = R$ and $I_ r(A) = 0$ if $r > \min (n, m)$. Then

1. $I_0(A) \supset I_1(A) \supset I_2(A) \supset \ldots$,

2. if $B$ is an $(n + n') \times m$ matrix, and $A$ is the first $n$ rows of $B$, then $I_{r + n'}(B) \subset I_ r(A)$,

3. if $C$ is an $n \times n$ matrix then $I_ r(CA) \subset I_ r(A)$.

4. If $A$ is a block matrix

$\left( \begin{matrix} A_1 & 0 \\ 0 & A_2 \end{matrix} \right)$

then $I_ r(A) = \sum _{r_1 + r_2 = r} I_{r_1}(A_1) I_{r_2}(A_2)$.

Proof. Omitted. (Hint: Use that a determinant can be computed by expanding along a column or a row.) $\square$

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