Suppose $F[x]$ is a field, and $p(x) \in F[x]$ be a polynomial. We want to quotient $F[x]$ with the ideal generated by $p(x)$.
While the commonly used notation to denote that $F[x]/p(x)$, isn't the notation kinda misleading because $p(x)$ is not an ideal ?
I think $F[x]/p(x)F[x]$ should be the correct notation since $p(x)F[x]$ is an ideal.