Prove Theorem 12.2.REFERENCE:Theorem 12.2 Uniqueness of

Step 1 of 2

Theorem 12.2 states that if a ring has a unity, then it is unique. If a ring element has a multiplicative inverse, then it is unique.

Let us consider a ring R with unity.

First, let us prove that unity is unique.

If possible, let us assume that R has two unity elements and .

Now, e is a unity in R and . Therefore, by definition of unity element,

                (i)

Again, is a unity element in R and . Therefore,

                 (ii)

From (i) and (ii), we have

It proves the uniqueness of the unity element in R.