Can you always define the sum of ideals?

Question

Let $R$ be a ring and $I_1, I_2$ be ideals. Then the sum of these ideals is defined as:

$I_1+I_2=\{s_1+s_2| s_1\in I_1\,\text{and}\, s_2\in I_2\}$

Is it always possible to define this sum of ideals, or just for a finite amount of ideals?

Let $\{I_k\}_{k\in K}$ be a family of ideals. In which case can you define $\sum_{k\in K} I_k$? Has $K$ to be finite, countable or can $K$ even be uncountable?

Since we are in a ring the addition is closed and always gives an element of $R$. So $\sum_{k\in K} I_k$ is always well defined.

Am I right?

Thanks in advance.

Answer 1

For every family $\{I_k\}_{k \in K}$, $\sum_k I_k$ is defined to be the ideal generated by $\cup_k I_k$. It is the same thing that the set of all finite sums $a_1+ \cdots + a_n$ where $a_i$ is in some $I_k$.