TITLE: Do Boolean rings always have a unit element?
QUESTION [9 upvotes]: Let $(B, +, \cdot)$ be a non-trivial ring with the property that every $x \in B$ satisfies $x \cdot x = x$. How does one prove that such a ring $(B, +, \cdot)$ must have a unit element $1_B$? (Or, in case this is not true in general, what is a counterexample?)

BTW, I'm looking for an elementary proof, not requiring anything more than the definition of a ring, the definition of $(B, +, \cdot)$, and, if necessary, the easily shown facts that $x + x = 0$ and $x\cdot y = y\cdot x,\,\forall\, x,y \in B$.

REPLY [5 votes]: In the ring $\prod_{i=1}^\infty\Bbb Z_2$, consider the ideal $\oplus_{i=1}^\infty\Bbb Z_2$. It is a subrng without identity.
(Ultimately I think this is isomorphic to Boris' example, but here the operations are clear.)