Definition clarification on ideals

Question

Suppose $P$ is the set of all subsets of a set $X$ and $P$ is a ring. Let $p$ be an element in $P$ (so that $p$ is a subset of $X$). What does it mean to say "an ideal generated by $p$"? And suppose there is some $q\in P$ then what does it mean to say "an ideal generated by (p,q)"? 

Answer 1

$P$ is a Boolean algebra (with $\cup = \lor$ and $\cap = \land$) and there is a notion of ideal there: an ideal $I$ of elements from $P$ obeys:

(1) $\forall p,q \in I : p \lor q \in I$

(2) $\forall p \in I, \forall q \in P: q \le p \rightarrow q \in I$

Which translates in your case as: an ideal is a collection of subsets closed under finite unions and subsets.

So the ideal generated by an element $p$ must contain all subsets of $p$, and then note that the collection of subsets of $p$ is closed under unions. So the ideal generated by $p$ equals $\{ q \mid q \subset p \}$.

The ideal generated by $p$ and $q$ must contain $p \cup q$ and all of its subsets, and one checks easily that it indeed is an ideal. So the ideal generated by $(p,q)$ equals $\{ (r \in P \mid r \subset p \cup q \}$

Answer 2

It could be that $P$ is considered as a Boolean ring (or algebra). In this case the ideals are the same as the order ideals of $P$. These are subsets of $P$ (collections of subsets of $X$) closed under finite unions and under intersections with arbitrary elements of $P$ (the latter menas they are lower sets: with every part of $X$ that is in the ideal, all smaller parts are also in it). An ideal generated by a finite number of sets in $P$ is the collection of all the subsets of their union; Henno Bransdma described what this means for ideals generated by one or two elements of $P$.