Is a multiplicative inverse a unit itself?

Question

Previously I've viewed units and multiplicative inverses to be mutually exclusive objects, but since every unit has a multiplicative inverse, then surely that unit acts as a multiplicative inverse for the original multiplicative inverse? So is it valid to say that every multiplicative inverse is also a unit itself?

Answer 1

Indeed, an element of a ring is a unit if and only if it is a multiplicative inverse. Namely, $u$ is a unit if and only if it is the multiplicative inverse of its own multiplicative inverse $u^{-1}$.