What does the ring $\mathbb{R}[x,y,z]/(x^2+y^2+z^2-1)$ look like? I know that in $\mathbb{R}[x,y,z]/(x^2+y^2+z^2-1)$ we have $x^2+y^2+z^2=1$ and my instinct would be to say that any element in the ring has the form $a_0+a_1x+a_2y+a_3z+a_4x^2+a_5y^2+a_6z^2$. Is that correct or could we lose the squared terms because of the above relation?
 A: I'm afraid your instinct isn't quite right - for example, $x^3$ is an element in your ring, which isn't of the form you described above.
The relation really says that you're allowed to convert $x^2 + y^2 + z^2$ into $1$, so you can e.g. always get rid of the $x^2$ term of any polynomial, or the $y^2$ term, or the $z^2$ term, or the constant term (which is perhaps the most "symmetrical" way of putting it into a standard form).
Another answer, from algebraic geometry, is that it's exactly the ring of polynomials in $\mathbb R[x,y,z]$, with two considered the same if they happen to define the same function on the unit sphere $x^2 + y^2 + z^2 = 1$. Equivalently, it's the ring of functions from the unit sphere to $\mathbb R$ that can be expressed by a polynomial.
However, neither of these are particularly satisfying answers to the question! Unfortunately the answer is that on some level you just have to deal with the ring as it is, there isn't a significantly simpler ring that it is isomorphic to (as far as I can see).
