order of quotient ring $\mathbb{Z}[x]/(x^2+x+1)(x^3+x+1)$

Question

I want to find order of quotient ring $\mathbb{Z}[x]/\langle(x^2+x+1)(x^3+x+1)\rangle$, Since the ideal is not prime. This is not integral domain. I am stuck. Similar problem is asked here, but to find order of $R/I$. This is different question. I feel order is infinity, but I cannot conclude

Answer 1

Let $R = \mathbb{Z}[x]/\langle(x^2+x+1)(x^3+x+1)\rangle$, and consider $n,m \in \mathbb{Z}$, $n\not=m$ as elements of $\mathbb{Z}[x]$. We have $n \not \equiv m \mod (x^2+x+1)(x^3+x+1)$ since otherwise $n-m \in \mathbb{Z}\setminus \{0\}$ would be a non-zero multiple of $(x^2+x+1)(x^3+x+1)$, and as such would have a higher degree in $x$.

This means that all elements of $\mathbb{Z}$ are different in $R$, so the cardinality of $R$ must be at least that of $\mathbb{Z}$, in other words the cardinality of $R$ is infinite.