Group of Units is Group

Theorem

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

Then the set of units of $\struct {R, +, \circ}$ forms a group under $\circ$.


Hence the justification for referring to the group of units of $\struct {R, +, \circ}$.


Proof

Follows directly from Invertible Elements of Monoid form Subgroup of Cancellable Elements.

$\blacksquare$


Sources

  • 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {IV}$: Rings and Fields: $21$. Rings and Integral Domains
  • 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): $\S 55.5$ Special types of ring and ring elements
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