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        riddles >> putnam exam (pure math) >> The Axioms of Algebraic Structures
        
(Message started by: peoplepower on Oct 30^th, 2012, 3:26pm)

Title: The Axioms of Algebraic Structures
Post by peoplepower on Oct 30^th, 2012, 3:26pm

Here are three problems. Two of which I know for a fact are well known, and the other I can only assume is well known.

Easiest: Let R be a ring (with identity) possibly with noncommutative addition. Prove that addition commutes anyway.
Easy: Let G be a finite set equipped with an operation (juxtaposition) such that left cancellation and right cancellation both hold. Prove that G is a group.
Easy-Medium: Let G be a set equipped with an operation (juxtaposition) such that there is a left identity and for every element there is a left inverse. Prove that G is a group.