solutions of polynomial with coefficients from Ring R

Question

Let $R$ be a ring and $p$ be a polynomial of degree $n$ with coefficients from $R$. Then is it true that $p$ can have at most $n$ roots in $R$?

Answer 1

No, it is not true. If $R=\Bbb Z_8$ the equation $X^2=1$ has four solutions.

Answer 2

If $R$ is an integral domain, then yes.

Otherwise (assuming this is a commutative, uital ring), if you take $a, b\in R\setminus \{0\}$ such that $a\neq b$ and $ab = 0$, then the polynomial $$ p = (x-a)(x-b) $$ has at least three roots $x = 0, x = a, x = b$.