Let be a ring and Let and Clearly is a multiplicatively closed subset of Let The claim is that To see this we define to be the inclusion map, i.e. for all We will show that the three conditions in the definition of a left qutient ring, mentioned in the previous section, are satisfied:

1) For any we have

2) Let where with If then and If let and See that

3) Since is the inclusion map, we have On the other hand if then and hence, since is an automorphism, we have if and only if

It’s even easier to prove that So is both the left and the right quotient ring of with respect to

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