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?
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2@reuns: No: an identity is an equation of two expressions built from variables and the operations which is true for all values you plug in for the variables from $\mathbb{R}$. – Eric Wofsey Jan 29 '20 at 03:19
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Ok, thus it is asking to list all the finite tree of product/sums representations of the polynomial $0\in \Bbb{Z}[x_1,\ldots,x_n]$ ? – reuns Jan 29 '20 at 03:22
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2@reuns: It's not asking you to list anything. It's asking for a finite set of identities in $\mathbb{R}$ which imply all the other identities in $\mathbb{R}$. – Eric Wofsey Jan 29 '20 at 03:25
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@EricWofsey ?? But it is the same. The axioms are just an algorithm generating the language, and the generative algorithm point of view is way more natural. – reuns Jan 29 '20 at 03:28
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@EricWofsey: thanks for pointing out my misreading of the present question. I've deleted my irrelevant comment. – Rob Arthan Jan 29 '20 at 20:07
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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.
Eric Wofsey
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