  Vanishing polynomials are polynomials over a ring which output $0$ for all
elements in the ring. In this paper, we study the ideal of vanishing
polynomials over specific types of rings, along with the closely related ring
of polynomial functions. In particular, we provide several results on
generating vanishing polynomials. We first analyze the ideal of vanishing
polynomial over $\mathbb{Z}_n$, the ring of the integers modulo $n$. We then
establish an isomorphism between the vanishing polynomials of a ring and the
vanishing polynomials of the constituent rings in its decomposition. Lastly, we
generalize our results to study the ideal of vanishing polynomials over
arbitrary commutative rings.
