Some basic elementary properties of a ring can be illustrated with help of following theorems and these properties are used in developing further concepts in rings and these properties are building of rings.
If is a ring, then for all are in .
(a) We know that
Since is a group under addition, applying 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) Actually to prove is a special case of forgoing article. However its proof is given as under:
Because is a consequence of the fact that in a group inverse of the inverse of an element is element itself.