[[Ring]]
# Group of units

For any [[ring]] $R$ there exists a **group of units** $R^\times$ or **multiplicative group** under ring multiplication, #m/def/ring
containing all [[Unit|units]].
This is clearly a group since it contains the multiplicative identity, is associative, and every element has an inverse.

## Properties

- For a [[division ring]] $R$, the multiplicative group $R^\times = R \setminus \{ 0 \}$ as a set
- [[Finite subgroup of the group of units of a field is cyclic]]

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#state/tidy | #lang/en | #SemBr