# Definition:Group of Units

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## Definition

### Group of Units of Monoid

Let $\struct {S, \circ}$ be a monoid.

Then the set $U_S$ of invertible elements of $\struct {S, \circ}$ can be referred to as the **group of units** of $\struct {S, \circ}$.

This can be denoted explicitly as $\struct {U_S, \circ}$.

### Group of Units of Ring

Let $\struct {R, +, \circ}$ be a ring with unity.

Then the set $U_R$ of units of $\struct {R, +, \circ}$ is called the **group of units** of $\struct {R, +, \circ}$.

This can be denoted explicitly as $\struct {U_R, \circ}$.