Is it possible to perform an MPC protocol to compute a product in a polynomial ring?
You mention a ring but you are only using the multiplication operation. So it's really just a protocol for a group operation that you want. If the group is abelian, then there is a simple classical protocol for $n$-party group product:


Each party performs $n$-out-of-$n$ multiplicative secret sharing of their input. In more detail, party $i$ has input $P_i$ and chooses random polynomials $P_{i,j}$ so that $\prod_j P_{i,j} = P_i$, then privately sends $P_{i,j}$ to party $j$.
Each party $j$ now has $\{ P_{i,j} \mid i \in [n] \}$. He/she computes their product $P'_j = \prod_i P_{i,j}$ and publicly announces $P'_j$.
If the group is abelian, then $\prod_j P'_j = \prod_{i,j} P_{i,j} = \prod_i P_i$, so everyone can compute the final answer from the public $P'_j$ values.


Furthermore, the view of any number of colluding semi-honest parties leaks no more than the final result.