In my problem, $R$ is a commutative ring with identity. Suppose $A$ is an n by n matrix with entries from R. Let $\operatorname{rk}(A)$ be the largest $m$ such that some $m$ by $m$ submatrix of $A$ has non-zero determinant (For A=0, put rk A=0.)
a) If $\operatorname{rk}(A)< n$ , show there is some non-zero n by $\operatorname{rk}(A)+1$ matrix B such that AB=0.
b) Hence prove that if $z_1, z_2, \cdots z_n$ are linearly independent elements of $R^{\ell}$ then $n\leq \ell.$
In class we have shown that the usual definitions for the determinant and adjugate matrix still work fine over general comm rings and some basic properties hold like the column/row expansion formulas for the determinant, and $\operatorname{adj}(A)A= A\operatorname{adj}(A)=\det(A)I_n.$
Thus, if $\operatorname{rk}(A)=n-1$ then a suitable matrix $B$ is $\operatorname{adj}(A).$ This satisfies $AB=0$ and $B$ is non-zero because its entries are (up to sign) determinants of n-1 by n-1 submatrices of $A$ and since rk A =n-1, at least one of those entries is non-zero. But this example fails if $\operatorname{rk}(A) < n-1$ and I haven't been able to make any progress for hours now. I also have no idea on how to do part b). Any help is greatly appreciated.