I'm looking for a general strategy for simplifying this sort of ring, or verification that I'm doing this correctly, and my logic is sound. I'd like to express it in terms of something more simple. Perhaps it is isomorphic to a finite field or a direct product of finite fields.
$R=\mathbb{Z}[x]/I$ where $I=( x^4+x^3+x^2+x+1 , x^6-x^5-x^4+3x^3-x^2-x+1 )$
My strategy so far has been to find other elements in the ideal like so, using a long division type strategy,
$(x^6 - x^5 - x^4 + 3 x^3 - x^2 - x + 1) - (x^4 + x^3 + x^2 + x + 1) (x^2 - 2 x)=4x^3+x+1 \in I$
Calculating further I find $x^2-3x+1$ and $3x^2+2x+3$ and $11$ are in the ideal. Thus we can consider the polynomials over $\mathbb{F_{11}}$. But the two quadratics above are not irreducible over that field, thus $R$ is not isomorphic to the field $\mathbb{F_{121}}$ but is isomorphic to the ring $\mathbb{F_{11}^2}$. Is this reasoning correct?