Rings

In a ring with identity $1$, an element $a$ is a unit if there exists an element $b$ such that $ab=ba=1$; $b$ is called the inverse of $a$. In ${\mathbb{Z}}_n$, an element $a$ is a unit exactly when $\gcd (a,n)=1$. A nonzero element $a$ is a zero divisor if there is a nonzero element $b$ with $ab=0$ or $ba=0$. A commutative ring with identity and no zero divisors is an integral domain.