Let $I,J$ be ideals of ring $R$. The product of two ideals is defined by $\sum_{\text{finite}}ab : a\in I,b\in J$.
My question is why do we take finite sums when defining the product of ideals?
For example: Consider $R=\mathbb Z$ ,$I= 5\mathbb Z$ and $J=7\mathbb Z$ then $IJ=\sum a_i b_j:a_i\in I,b_j\in J=\sum (5k)(7\ell):k,\ell \in \mathbb Z=\sum 35(k\ell)$
Even though $I$ and $J$ have infinite elements, why can't we add more elements from $I$ and $J$ in $\Sigma$?