Suppose there are two ideals $I,J \in \mathbb{C}[x_1,\dots,x_k]$ and two sets of generating polynomials $\langle f_1, \dots, f_s\rangle$, $\langle g_1, \dots, g_t\rangle$. Now I want to describe $I + J$ with a set of generating polynomials. Is
$$\langle f_1, \dots, f_s, g_1, \dots, g_t\rangle$$
a valid generating system?
Best Answer
Yes, you are correct.
One way to see this is to note that $I+J$ is the smallest ideal that contains both $I$ and $J$; indeed, $I\subseteq I+J$ by taking $y=0$, and $J\subseteq I+J$ by taking $x=0$. And if $K$ is an ideal that contains $I$ and $J$, then it contains $x+y$ for every $x\in I$ and $y\in J$, so $I+J\subseteq K$. Finally, $I+J$ is an ideal: it is a subgroup, and if $r\in R$, then $r(x+y) = (rx)+(ry)\in I+J$ since $I$ and $J$ are ideals, and $(x+y)r = (xr)+(yr)\in I+J$ because $I$ and $J$ are ideals.
In particular, if $I=\langle f_1,\ldots,f_s\rangle$ and $J=\langle g_1,\ldots,g_t\rangle$, then $f_i,g_j\in I+J$ for all $I$ and $J$, so $\langle f_1,\ldots,f_s,g_1,\ldots,g_t\rangle\subseteq I+J$; and since $\langle f_1,\ldots,f_s,g_1,\ldots,g_t\rangle$ is an ideal that contains both $I$ and $J$, then $I+J\subseteq \langle f_1,\ldots f_s,g_1,\ldots,g_t\rangle$, proving the equality.