# The Unapologetic Mathematician

## Boolean Rings

A “Boolean ring” is a commutative ring $R$ with the additional property that each and every element is idempotent. That is, for any $r\in R$ we have $r^2=r$. An immediate consequence of this axiom is that $r+r=0$, since we can calculate \displaystyle\begin{aligned}0&=(r+r)-(r+r)\\&=(r+r)^2-(r+r)\\&=r^2+r^2+r^2+r^2-(r+r)\\&=r+r+r+r-(r+r)\\&=r+r\end{aligned}

The typical example we care about in the measure-theoretic context is a ring of subsets of some set $X$, with the operation $E\Delta F$ for addition and $E\cap F$ for multiplication. You should check that these operations satisfy the axioms of a Boolean ring. Since this is our main motivation, we will just consistently use $\Delta$ and $\cap$ to denote addition and multiplication in Boolean rings, whether they arise from a measure theoretic context or not. From here it looks a lot like set theory, but keep in mind that the objects we’re looking at may have nothing to do with sets.

We can use these operations to define the other common set-theoretic operations. Indeed $\displaystyle E\cup F=(E\Delta F)\Delta(E\cap F)$

and $\displaystyle E\setminus F=E\Delta(E\cap F)$

and we can then define orders in the usual manner: $E\subseteq F\Leftrightarrow E\cap F=E$.

As usual, the union of two elements is the “smallest” (with respect to this order) element above both of them, and the intersection of two elements is the “largest” element below both of them. The same goes for any finite number of elements, but if we try to move to an infinite number of elements there is no guarantee that there is any element above or below all of them, much less that such an element is unique. A “Boolean $\sigma$-ring” is a Boolean ring so that every countably infinite set of elements has a union. In this case, it is immediately true that any countably infinite set of elements has an intersection as well. The typical example, of course, is a $\sigma$-ring of subsets of a set $X$.

A “Boolean algebra” is a Boolean ring for which there is some element $X\neq0$ so that $E\subseteq X$ for all elements $E$. A “Boolean $\sigma$-algebra” is both a Boolean $\sigma$-ring and a Boolean algebra.

In the obvious way we have a full subcategory $\mathcal{B}oolean$ of the category of rings. It contains full subcategories of Boolean $\sigma$-rings, Boolean algebras, and Boolean $\sigma$-algebras.

August 4, 2010 Posted by | Algebra, Ring theory | 11 Comments