I was given the following question:

A is the set Q of the rational numbers, and the operations are$\displaystyle \oplus$ and $\displaystyle \otimes$ defined as follows:

a$\displaystyle \oplus$b = a + b + 1

a$\displaystyle \otimes$b = ab + a + b

I have proved that it is a ring. So all I have to do is..

Prove that A is commutative

a$\displaystyle \otimes$b = ab + a + b

b$\displaystyle \otimes$a = ba + b + a

ab + a + b = ba + b + a

ab + a + (b - b) - a = ba + b + (a - a) - b

ab + a + 0b - a = ba + b + a0 - b

ab + a + 0 - a = ba + b + 0 - b

ab + a - a = ba + b - b

ab + (a - a) = ba + (b - b)

ab + a0 = ba + b0

ab + 0 = ba + 0

ab = ba

I think I am off to the wrong start if someone can correct me that would be great

Prove that A has unity

Prove that every non zero element of A is invertible