My professor gave us this example on modules (he started this by saying what is a basis? what is the meaning of linearly independent?):
$R = k[x,y,z]$ where $k$ is a field and $I = xyR + yz R + xz R$. Take $u = xy, v = yz, w = xz$. Then $I = uR + vR + wR.$ So, $$zu - xv = 0, xv - yw = 0, zu - yw = 0.$$ Hence $$\begin{bmatrix} z & -x & 0\\ 0 & x & -y\\ z & 0 & -y \end{bmatrix}\begin{bmatrix} u \\ v \\ w \end{bmatrix} = \begin{bmatrix} 0\\ 0\\ 0 \end{bmatrix}, $$
$\det A = 0.$
The span of $u,v,w$ is $Ru + Rv + Rw = I$ so $I$ can not be spanned by $2$ elements.
Upshot: $I \ncong_R R^2 = R(1,0) \oplus R(0,1)$
And I am for my life do not understand the idea my professor wanted to convey, could someone help me in trying to understand my professor's mind?