Let $R$ be a ring. Let $I = \langle x + y \rangle$, $J = \langle x - y \rangle$ be ideals of $R[x,y]$.
What's $I + J$ in this case? By definition $I + J = \{ i + j \mid i \in I, j \in J \}$. My first thought was that $I + J = \langle 2x \rangle$ but since $x(x+y) \in I$ and $y(x-y) \in J$ and $x(x+y) + y(x-y) = (x-y)^2 \notin \langle 2x \rangle$ this must be wrong. Is there a formula for $I + J$ if $I,J$ are generated by some set of elements?