What characterizes the equivalence classes of the quotient ring, P(N)/Fin(N)? Let P(N) be the powerset of the natural numbers.  Let Fin(N) be the collection of all finite subsets of N. Then (P(N),symmetric difference, intersection) is a ring. I am taking my first course in ring theory (so I am not sure if I am using the correct terminology). I think I have made an interesting observation about the cosets of Fin(N).  My claim is that for subsets A and B in P(N),  A + Fin(N) = B + Fin(N) implies that there is a positive integer m such that for all n>=m, n is an element of BOTH A and B or n is in NEITHER A nor B.  In other words, if we compare A and B we eventually get to a point where they contain the same integers.  Does anyone agree/disagree?  
One reason I think this is interesting is that it illustrates a 1-1 correspondence between the cosets of Fin(N) (which themselves contain an infinite number of elements) and P(N).
 A: Yes. The collection of finite subsets forms an ideal (both ring theoretic and set theoretic). And since addition is an involution ($A+A=0$ for all $A$), it means that you in fact defined the usual quotient by an ideal with $A\equiv B\iff A-B=A+B\in\mathrm{Fin}$.
So two equivalence classes, are the same if and only if $A+B\in\mathrm{Fin}$, namely $A\mathop{\triangle}B$ is a finite set. So it means that $A$ and $B$ are equal after some initial segment.
(Interestingly, it does not quite illustrates a 1-1 correspondence, since in some models of $\sf ZF+\lnot AC$, there is no bijection between $\mathcal P(\Bbb N)$ and $\mathcal P(\Bbb N)/\mathrm{Fin}$.)
(Other footnotes may include that what is interesting is that you can find a sequence of order type $\omega_1+\omega_1^*$ in the order inherited by the quotient. Namely, there is a sequence $A_\alpha$ and $B_\alpha$ for $\alpha<\omega_1$, such that $A_\alpha$ form an increasing sequence in $\mathcal P(\Bbb N)/\mathrm{Fin}$, the $B_\alpha$ form a decreasing sequence, and each $A_\alpha$ is smaller than all the $B_\beta$'s .)
