is there any set of matrices, numbers or objects that satisfy the following?
1) $AB=0$ where A and B are different objects inside the set
2) they commute.
is there any set of matrices, numbers or objects that satisfy the following?
1) $AB=0$ where A and B are different objects inside the set
2) they commute.
if $AB=0$ and they commute, then shouldn't $BA=0$. Anyways, a set of matrices which satisfies your conditions would be a set of projection matrices to orthogonal subspaces. Let us divide $R^{N}$ into orthogonal subspaces $V_1$, $V_2$,...,$V_K$ with dimensions $n_1,n_2,\dots,n_K$ respectively such that $\sum_{i=1}^{K}n_i\leq N$. Then the projection matrices for each of them $P_1,P_2,\ldots,P_K$ respectively will form a set of matrices which satisfies your conditions.
$2\times3=0$ in ${\bf Z}/6{\bf Z}$, and that ring is commutative.