 Nonunit

A nonunit is an element of a ring with no two-sided multiplicative inverse in that ring. For example, $2$ is a nonunit in the integers, even though multiplication by $2$ obeys the cancellation law. Thus being a nonunit does not imply being a zero divisor.

The set of nonunits need not be an ideal. In $\mathbb{Z}$, both $2$ and $-3$ are nonunits, but their sum $-1$ is a unit. In a commutative local ring, by contrast, the nonunits form its unique maximal ideal.