Union of Chain of Ideals I'm writing a project in a "Rings and Modules" course, and I've come across the following proposition, stated without proof:
Proposition 1.2.
In a commutative ring R , the product of ideals is commutative and associative, and distributes sums and unions of chains.
Generally, whenever one of these occurs I try to prove it myself. However in this case I'm struggling to understand what the proposition is even stating. I've already proven that the product distributes sums of ideals, but I can't find a nice definition of what a chain is.
As far as I can gather, the question is saying show that
$A(B\cup C)=A\cup C + B\cup C)$ where A is an ideal of R, and $B,C$ are chains. However without knowing what a chain is I can't really progress. I know that a chain must be an ideal, else we couldn't have ideal multiplication between A and B, for example. So I assume that 
$$I_1\subseteq I_2 \subseteq ... = B$$
not that I find that an acceptable definition of a chain at all. Then would a union of chains be $\cup_i I_i=B$ or would it be $B \cup C$?
Well that's left me really confused anyway. If anyone could clarify I'd be greatly appreciative. Thanks for any replies!

I would interpret it as saying 
$I \sum_{\alpha\in \mathcal{A}} J_\alpha = \sum_\alpha IJ_\alpha$
whenever $\{J_\alpha : \alpha \in \mathcal{A}\}$ is a set of ideals,
(product distributes over sums)
and
$I \cdot \bigcup_\alpha J_\alpha = \bigcup_\alpha IJ_\alpha$
whenever $\{J_\alpha : \alpha \in \mathcal{A}\}$ is a chain of ideals
(product distributes over chains).
A set of ideals $\{J_\alpha : \alpha \in \mathcal{A}\}$ is a chain of ideals if for all $\alpha, \beta \in \mathcal{A}$, either $J_\alpha \subseteq J_\beta$ or $J_\beta \subseteq J_\alpha$.
