Thread: Abstract algebra - associative commutative binary operations?

1. Abstract algebra - associative commutative binary operations?

Show that *,defined on Q (where Q is a set of rational numbers) by

a * b = a + b + 3ab

is a commutative binary operation.Is is associative?
Ive answered this part but i dont know how to answer part b

Determine the identity element admitted by * and show that it is unique.Show that with respect to this identity the inverse a(^ -1) of a is given by

a(^ -1) = ((-a)/(1+3a))

Give an element a C Q which has no inverse with respect to *.
(where C means contained, Q is a set of rational numbers)

2. If we denote the identity by e, then for any a in Q:
we have a*e=a+e+3ae=a
thus e=0.
$a*a^{-1}=e=0$ gives the formula of the inverse of a.
a has no inverse iff 1+3a=0.

3. Originally Posted by Shanks
$a*a^{-1}=e=0$ gives the formula of the inverse of a.
a has no inverse iff 1+3a=0.