Given a set $X$, we can construct the Boolean ring whose elements are the power set of $X$. The multiplication therein is intersection, and the addition is symmetric difference. I am interested in finding information on the study of such rings, for instance, information about the spectrum of such a ring. In that case, I am specifically interested in trying to find a systematic description of what the localization of such a ring at an object (i.e. a subset of $X$) would look like. Doing some basic calculations, strange things seem to happen. For instance, if $X$ is the natural numbers, and we localize at (i.e. invert all powers of) the set $\{1,2\}$, we get constructions like $\{1,2\}+1/\{1,2\}$, which is an object that when multiplied by (intersected with) the set $\{1,2\}$, yields $X-\{1,2\}$, which is pretty wild. This is a pretty algebraically simple subject, but I don't recall ever seeing anything written down about it.
Thanks for any help!