# Abstract algebra proofs

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#### Solution Preview

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1. One of the defining rules of a ring tells us the statement is true when we have two elements in the bracket. That is, (please see the attached file).

The trick is to recognize that we can insert extra brackets to transform the bracket into a small of a smaller number of elements, then apply induction. For example, for (please see the attached file) ,(please see the attached file).

That's the idea. For a formal proof we use induction. Assume the rule has been proved for n or less elements in the bracket. Then (please see the attached file)

It's worth learning this technique carefully because it is used repeatedly. For example, associativity of a product of any number of elements follows in a very similar way from the associativity of 3 elements.

2. is by definition with a making m appearances. Thus (please see the attached ...

#### Solution Summary

This solution provides examples of abstract algebra proofs regarding distributive law, commutative rings, integral domains, and elements of rings.