Regular element

Let $R$ be a ring. An element $a\in R$ is left regular if $ax=ay$ implies $x=y$ for all $x,y\in R$, and right regular if $xa=ya$ implies $x=y$ for all $x,y\in R$. It is regular if it is both left and right regular. In a commutative ring, this is equivalent to saying $a$ is not a zero divisor , i.e. $ax=0$ implies $x=0$.

Regular elements are precisely those with trivial annihilator (in the commutative case), and they are the multiplicative set used to form total rings of fractions and localizations.

Examples:

• In $\mathbb Z$, every nonzero integer is regular.
• In $\mathbb Z/6\mathbb Z$, the class of $5$ is regular, while the class of $2$ is not.
• In $k[x,y]/(xy)$, the class of $x$ is not regular.