Ring

Group of units

For any ring 𝑅 there exists a group of units 𝑅× or multiplicative group under ring multiplication, ring containing all 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× =𝑅 ∖{0} as a set
• Finite subgroup of the group of units of a field is cyclic

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