$Z[x]$ is a ring. $I$ is an ideal of polynomials with even coefficients. Need to prove it is maxximal.
So I take another ideal $J$ and I know there is in J a polynomial with at least one odd coefficient. And because J is an ideal, it is closed under substraction, so I can take a similiar polynomial that in I (and therefore it will be from J eather), with coefficient bigger by 1 (and of course even) of the odd coeficient in the polynomial who is in J, but not in I.
Now by substraction of that two polinomials, I'll get $x^{i}\in J$. What I really want to show is that $1\in J$. Now can I say that in $x^{-i}\in Z[x]$ and therefore $x^{i}*x^{-i}\in J$ ? If not, how from the fact that there is polynomial in J with at least one odd coefficient, I can conclude that $1\in J$ or $J=R$?