[Soft question out of curiosity]
While reading about idempotents of a ring, I used $\mathbb Z_n$ as a convenient example. In visualizing the ring's structure, I was intrigued by strange symmetries in squared elements:
It seems that, in $\mathbb Z_n$, if n is even, the 'trajectories' of all squared elements are symmetrically arranged, and in all cases, the idempotent elements of the ring are symmetrically arranged.
(examples in $\mathbb Z_6$, black arrows indicate squaring, red indicate multiplication of two different elements)
this seems to be related to the fact that the ring is a direct sum of fields ($\mathbb Z_p$), as subrings... the symmetry being an artifact of the ring containing 'redundancies' from being a direct sum.
Does anyone know why this occurs or if there is anything to learn from these symmetries? (if you know several idempotents, you can find the rest by symmetry.. I suppose).