TITLE: Characterization of the field $\mathbb{Z}/2\mathbb{Z}$
QUESTION [9 upvotes]: Let $R \neq 0$ be a ring which may not be commutative and may not have an identity.
Suppose $R$ satisfies the following conditions.
1) $a^2 = a$ for every element $a$ of $R$.
2) $ab \neq 0$ whenever $a \neq 0$ and $b\neq 0$.
Is $R$ isomorphic to the field $\mathbb{Z}/2\mathbb{Z}$?

REPLY [2 votes]: Let $a \neq 0$ be an element of $R$.
It suffices to show that $a$ is an identity.
So we would like to show first that $ab = b$ for any element $b$ of $R$.
It suffices to show that $a(ab) = ab$ thanks to the condition 2).
But this is obvious thanks to the condition 1).
Similiarly we can show that $ba = b$.