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 .
Proof: (a) We know that
Since is a group under addition, applying right cancellation law,
Applying right cancellation law for addition, we get i.e.
Proof: (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
Proof: (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.