Below is a proposition about rings, and a short proof in which the use of the axioms and other properties of rings is not made explicit. Proposition. Let R be a ring. For any two elements a and b of R, it holds that -(ab) = (-a) b. Proof. We have -(ab) + ab= 0 and (-a) b + ab= 0. Therefore -(ab) = (-a)b. Which property is used to prove -(ab) + ab=0? This property and two more are used to prove (-a) b + ab=0. What are the other two? Put them in the order you would use them when simplifying (-a) b + ab to 0. First then Which property is used to finish the proof? Please look at the lecture notes to remind yourself e.g. what "Proposition 3.13" is. commutative law for + associative law for • definition of inverse for + identity law for + identity law for. definition of inverse for. > associative law for + Proposition 3.13 cancellation property distributive law

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