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When people talk about algebraic identities, such as in A Collection of Algebraic Identities, are those variables appearing in them varying in $\mathbb{R}$, $\mathbb{C}$ or some even more general set? I am particularly interested in the last possibility, i.e. what is the most general set where algebraic identities hold?

Are algebraic identities subjects studied in number theory or abstract algebra?

Thanks and regards!

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    The question seems a bit vague. In a nutshell, Algebraic identities often means polynomial identities. This involves products, sums (and sometimes multiplication by a scalar, or even inverses). So you could hope to have some identities in any structure where those operations make sense : rings (sum, product), algebras (sum, product, multiplication by a scalar), fields (sum, product, inversion) etc.2011-05-08

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Where the variables come from can only be ascertained from the context in which they appear. Your link gives a huge number of identities, and I'm not going to go through the whole site to see the context for each one. Here's one of them: $n^2+(n+1)^2+(n^2+n)^2=(n^2+n+1)^2$ It's clear from the context that the author has it in mind that $n$ is an integer. This particular identity certainly holds in any ring that has a $1$ in it.

Algebraic identities are studied in Number Theory and in abstract algebra.

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The identities you reference are almost all comprised of (multivariate) polynomials with integer coefficients. As such they remain true over any commutative ring, whether interpreted formally (equal iff coefficients are equal) or functionally (equal as functions on an infinite coefficient ring).

Beware that equivalence between formal and functional polynomials may fail over finite rings, e.g. $\rm\: x^p = x\:$ as functions on $\rm\:\mathbb Z/p\ =\:$ integers $\rm\:(mod\ p)\:,\ $ since $\rm\ a^p \equiv a\ (mod\ p)\ $ for all $\rm\ a\in\mathbb Z/p\ $ by Fermat's little Theorem. However this equality is not true when considered as formal polynomials over $\rm\mathbb Z/p$ since, by definition, formal polynomials are equal iff their coefficient sequences are equal; equivalently, subtracting, a formal polynomial is zero iff all of its coefficients are zero.

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    @Tim: I've expanded the answer. Please feel free to ask a more specific question if something still is not clear.2011-05-08