Prove that the only boolean ring that is an integral domain is .
Let be a boolean ring which is an integral domain. If is nonzero, then , and by the cancellation law, . By §7.1 #11, or . Note also that , so that . Additively, , and in fact and , so that “is” . (The book hasn’t introduced ring isomorphisms yet.)
Alternate proof: Suppose is not 0 or 1. Then and are nonzero, but , so that is a zero divisor and we have a contradiction. Thus .