Arbitrary product of ideals

Question

For a commutative ring $R$ with 1, let $\{I_i\}_i$ be an infinite family of ideals of $R$. I think that we can define the arbitrary product for $I_i's$ as follows:

$\prod_i I_i=< a_{i_1}a_{i_2}...a_{i_n}>$ Where $n\in\mathbb{N}$, ${i_1}, {i_2},...,{i_n}\in I$ and $a_{i_j}\in I_{i_j}$ Is there any result about this definition?

Answer 1

This definition is not nice, by your definition, since for each $k\in I$ and $a_k\in I_k$, we have $a_k\in \prod_i I_i$. Thus, $\prod_i I_i$ is exactly the sum of $I_i$.