It's likely this is trivial, but I can't figure it out. My book constructs the following quotient ring: $\mathbb{Z}_2[x]/\langle{x^3+x+1}\rangle$
It then makes the following computation as an example: $$(x^2+x+1+\langle{x^3+x+1}\rangle)+(x^2+1+\langle{x^3+x+1}\rangle) \\ = x+\langle{x^3+x+1}\rangle$$
How'd they get this? I'm getting: $2x^2+x+1+\langle{x^3+x+1}\rangle$. More confusingly, $2x^2+x+1$ has degree less than the generator -- how is it getting absorbed into the ideal at all?
Note that $(x^2+x+1)+(x^2+1)=2x^2+x+2=x$, since we're working on $\mathbb{Z}_2[x]$.