# defining binary operation on $\mathbb{R}$

Exercise 3, page 13 from Golan's book ("The Linear Algebra a Beginning Graduate Student Ought to Know"): Define a new operation $\circ$ on $\mathbb{R}$ by setting $a\circ b= a^{3}b.$ Show that $\mathbb{R}$, on which we have the usual addition and this new operation as multiplication, satisfies all of the axioms of a field with the exception of one.

My work: The new operation is not commutative and there is no identity of multiplication. If there exists $x\in \mathbb{R}$ such that $a\circ x= a = x\circ a$, then $a^{3}x=a=x^{3}a$. If $a\neq 0$ then $x=1$ and the first equality would gives us $a=\pm 1$.

Is the exercise wrong or I am doing something wrong?

Certainly $1$ is a left identity. I don't know what book you're referring to, but it might only require left identity (which together with commutativity gives right identity. This is silly though). –  Qiaochu Yuan Jan 25 '12 at 21:14