2. Boolean rings

Definition. A Boolean ring is a ring in which every element is idempotent.

Examples.

1. ${\mbox{\bf Z}} _ 2$ as a ring.

   
   
   

2. ${\mbox{\bf Z}}_2^n$ as a ring ($n \geq 1$).

   
   
   

3. Pow$(X)$, made into a ring by letting multiplication be $\cap$, addition be the symmetric difference $A \Delta B = A \backslash B \cup B \backslash A$, and 0 be the empty set.

   
   
   

4. For an infinite set $X$, the subring of Pow$(X)$ consisting of the finite subsets of $X$.

   
   
   

5. For any Boolean algebra $B = \langle B,\vee, \wedge, ', 0, 1 \rangle$, the ring $\langle B, +, \cdot, 0 \rangle$ obtained by defining $xy$ to be $x \wedge y$ and $x+y$ to be $(x \wedge y') \vee (y \wedge x')$, the Boolean-algebra analogue of the symmetric difference.

All these examples except . are Boolean rings with 1.

Proposition 1. Any Boolean ring is of characteristic two (i.e., obeys $x+x = 0$ for all $x$).

Proposition 2. Any Boolean ring is commutative.

Proposition 3. Any Boolean ring with 1 can be made into a Boolean algebra by defining $x \wedge y = xy$, $x \vee y = x + y + xy$, and $x' = 1-x$.

Proposition 4. For a Boolean algebra $B$, a subset is a lattice ideal if and only if it is a ring ideal with respect to the resulting ring structure.