Problem

Source: Romanian National Olympiad 1998 - Grade 12 - Problem 3

Tags: abstract algebra, Rings



A ring $A$ is called Boolean if $x^2 = x$ for every $x \in A$. Prove that: a) A finite set $A$ with $n \geq 2$ elements can be equipped with the structure of a Boolean ring if and only if $n = 2^k$ for some positive integer $k$. b) The set of nonnegative integers can be equipped with the structure of a Boolean ring.