Polynomial Functions Over Finite Rings

In this paper, we explore properties of the ring of polynomials P(R, R) over a finite ring R. We examine the kernel of the evaluation map E to find #P(R,R), first for when nilpotency index a is at most residue cardinality q, and next when a=q+1 and a>q. We identify criteria for when all ideals of P(R,R) are principal and look at concrete examples, and we put an R-module structure on P(R,R). Finally, we briefly examine these same questions for the ring P(R^N, R).