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.