This is a homework problem:
Consider the polynomial ring $R=\mathbb Z_2[x_0,x_1,\dots,x_n]$. Let $I=\langle x_0x_1\cdots x_n\rangle$ and $J=\langle x_0+x_1,x_0+x_2,\dots,x_0+x_n\rangle$. Find $I+J$. Hence find $R/(I+J)$.
I first tried thinking about a more general statement. If $R$ is a commutative ring with unity and $I=\langle f_1,\dots,f_i\rangle$ , $J=\langle g_1,\dots, g_j\rangle$ are ideals of $R$ then what can we say about the generators of $I+J$ in terms of the generators of $I$ and $J$? But even in the case if both are principal - $I=\langle f\rangle$ and $J=\langle g\rangle$ I am not sure how to proceed. I am convinced it is not $\langle a+b\rangle$. How would I proceed to tackle such a problem?