Definition 2.8.15.  Let R be a ring, a ∈ R. We define:
a is a unit  :⇔  ∃ b ∈ R : a ⋅ b = b ⋅ a = 1R  ⇔  ∃! b ∈ R : a ⋅ b = b ⋅ a = 1R
Equivalence.  No proof.
References.