# Elementary Properties of Rings

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.

__Theorem__**: **

If is a ring, then for all are in .

**(a)**

**(b)**

**(c)**

__Proof__**: **

**(a)** We know that

Since is a group under addition, applying the right cancellation law,

Similarly,

Applying right cancellation law for addition, we get i.e.

Thus

**(b)** To prove that we should show that

We know that because with the above result **(a)**

Similarly, to show , we must show that

But

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.