I need to find the dimension of the basis for $V=R[x]/(x^3-x)$ - I don't need to find the basis itself just its dimension but it would be nice to find it as well.I think that $V$ is the set of all polynomials in the form of $ v(x) + q(x)(x^3-x) $ for some $ q(x) \in R[x] $ where $v(x)$ is the remainder of the division of a polynomial in $R[x] $ with $(x^3-x)$. But here i'm stuck. My initiation is that $ q(x)(x^3-x) $ is the zero vector for $V$ and I need to find the basis for the vectors not in $ q(x)(x^3-x) $ but I'm not sure if $ q(x)(x^3-x) $ includes all polynomials of order 3 or above and if yes how to prove it.
Thanks!