Question

Let R be a ring possibly without an identity. Show that if R has a unique left identity e, then e is also a right identity

Answer 1

Suppose e ∈ R is a unique left identity for R. Then for any a, c ∈ R,
(e + ae − a)c = c + ac − ac = c.
Therefore, e + ae − a = e, which implies ae = a (for any a ∈ R).