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An undergraduate course in Abstract Algebra Course notes for ...

iv) a0 0 = a0 (Axiom (i))= a(0 0) (Axiom (ii))= a0 a0 (Axiom (vi)). ... By part (i) above it follows that a0 = 0. The proof that 0a = 0 is similar.
www.maths.usyd.edu.au/u/bobh/UoS/rfwhole.pdf