Finite equational basis for the identities of the field of real numbers

Question

Consider the structure $(\mathbb{R},+,-,\times,0,1)$, where the $-$ is the unary additive inverse function, not binary subtraction. Can someone exhibit a finite set of identities that can be used to derive all the rest of the identities of that algebraic structure? Or is there no finite basis for that structure?

Answer 1

Sure: the axioms of commutative rings are enough. Indeed, this follows immediately from the fact that the free commutative ring on any finite set embeds in $\mathbb{R}$ (just send the free generators to algebraically independent elements), so any identity which is true in $\mathbb{R}$ is also true in every commutative ring.