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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?

Thanks in advance!

PS.Please, correct any mistake I made.

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    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).2012-01-25
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    There are many different ways of specifying the axioms of a field, so it is difficult to say. One could state "multiplication is commutative" as one axiom, and then only require multiplicative identities and inverses on *one side* in the other axioms (just like we can define a group with axioms that *explicitly* require the identity and the inverses to be two-sided, or we can define a group with axioms that only require a left identity and left inverses). So you'd need to look at the axioms very carefully to see how they are stated, in order to decide which one(s) are not satisfied.2012-01-25
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    Can you post the title of the textbook in addition to the author? I found this: "The Linear Algebra a Beginning Graduate Student Ought to Know"; is this the one?2012-01-25
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    Hello Srivatsan! That is the book! I will give the reference next time. Thanks for point it out.2012-01-25
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    Note that the axiom on page 6 does not require the identity to be 2-sided. But it does require a right identity, and that doesn't hold here either. So you may have had a minor technical oversight, but you're still correct that there is more than one property missing.2012-01-25

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