We have (a, b) * (c, d) = (ad + bc, bd) for (a, b) (c, d) ∈ A
Let (p, q) be the identity element of (A, *)
∴ For (a, b) ∈ A, we have
(a, b) * (p, q) = (a, b) = (p, q) * (a, b)
⇒ (aq + bp, bq) = (a, b) = (pb + qa, qb)
⇒ aq + bp = α and bq = b
Solving, we get p = 0, q = 1
Since 0 N, (0, 1) A
∴ (A, *) has no identity element. Remark . Since identity element does not exist in the above example, the concept of inverse of an element is not defined in the set A.