For an algebraic variety $V$, denote its ring of regular functions by ${\cal O}(V)$. The *Kahler differentials* of $V$ are the quotient of the kernel $M$ of the multiplication map
$$
m: {\cal O}(V) \otimes {\cal O}(V)\to {\cal O}(V)
$$
by the ideal $M^2$.

What can be said about the maximal proper submodules of $M$?

Is there any sense/specific-case in which the submodule $M^2$ is maximal?

I am particularly interested in the homogeneous variety case, specifically the flag variety case. For example, is $M^2$ a maximal right $G$-invariant proper submodule when $V$ a a $G$-homogeneous variety.

affine. 2) $M$ is not a ring, therefore it does not make sense to talk about maximal ideals of $M$. Perhaps you mean maximal proper submodules? – Martin Brandenburg Oct 13 '11 at 17:47