If you have a finite ring $R$ that is [von Neumann regular](https://en.wikipedia.org/wiki/Von_Neumann_regular_ring), then $[0]=N(R)$ or more generally if you have a von Neumann regular ring $R$ such that each element has a power that belongs to a subsemigroup which is a group.  This follows from a result on semigroups in http://arxiv.org/abs/math/0605698, see also http://arxiv.org/pdf/0709.4341.pdf.  Maybe you can use some of the ideas with the ring property to get further.