Some basic elementary properties of a ring can be illustrated with the help of the following theorem, and these properties are used to further develop and build concepts on rings.
If is a ring, then for all are in .
(a) We know that
Since is a group under addition, applying the right cancellation law,
Applying right cancellation law for addition, we get i.e.
(b) To prove that we should show that
We know that because with the above result (a)
Similarly, to show , we must show that
hence the result.
(c) Proving is a special case of forgoing the article. However its proof is given as:
This is because is a consequence of the fact that in a group, the inverse of the inverse of an element is the element itself.