Exercise 8, § 9

A ring A is called a Boolean ring if each of its elements is idempotent (in other words, if $x^2 = x$ for all $x \in A$). (a) In the set $\mathscr{P}(E)$ of subsets of a set E, show that a Boolean ring structure is defined by setting $AB = A \cap B$ and $A + B = (A \cap \complement B) \cup (B \cap \complement A)$. This ring is isomorphic to the ring $K^E$ of mappings of E into the ring $K = \mathbf{Z}/(2)$ of integers modulo 2 (consider for each subset $X \subset E$ its "characteristic function" $\phi_X$ such that $\phi_X(x) = 1$ if $x \in X$, $\phi_X(x) = 0$ if $x \notin X$). (b) Every Boolean ring A is commutative and such that $x + x = 0$ for $x \in A$ (write $x + x$ as an idempotent and then $x + y$ as an idempotent). (c) If a Boolean ring A contains no divisor of 0, it is reduced to 0 or is isomorphic to $\mathbf{Z}/(2)$ (if $x$ and $y$ are any two elements of A, show that $xy(x + y) = 0$). Deduce that in a Boolean ring every prime ideal is maximal. (d) In a Boolean ring A, every ideal $a \neq A$ is the intersection of the prime ideals containing a (apply Exercise 7(c)). Deduce that every irreducible ideal is maximal (in other words, that the notions of irreducible ideal, prime ideal and maximal ideal coincide in a Boolean ring).

(e) Show that every Boolean ring is isomorphic to a subring of a product ring $K^E$, where $K = \mathbf{Z}/(2)$ (use (d)).

(f) If $p_i$ ($1 \leq i \leq n$) are $n$ distinct maximal ideals in a Boolean ring $A$ and $a = \bigcap_{1 \leq i \leq n} p_i$, show that the quotient ring $A/a$ is isomorphic to the product ring $K^n$. Deduce that every finite Boolean ring is of the form $K^n$.

(g) In a Boolean ring $A$, the relation $xy = x$ is an order relation; if it is denoted by $x \leq y$, show that with this relation $A$ is a distributive lattice (Set Theory, III, § 1, Exercise 16), has a least element $\alpha$ and that, for every ordered pair $(x, y)$ of elements such that $x \leq y$, there exists an element $d(x, y)$ such that $\inf(x, d(x, y)) = \alpha$, $\sup(x, d(x, y)) = y$. Conversely, if an ordered set $A$ has these three properties, show that the laws of composition $xy = \inf(x, y)$ and $x + y = d(\sup(x, y), \inf(x, y))$ define a Boolean ring structure on $A$.