Dear Shizhuo, here is a another proof that the two rings are not isomorphic.

By extending the scalars to $\mathbb C$ it suffices to prove that $SL(n,  \mathbb C)$ is not isomorphic  to affine space $\mathbb A^{n^2-1} _\mathbb C$. But these spaces are not  homeomorphic when endowed with their classical topology. Indeed affine space has no cohomology (it is homotopic to a point), whereas  the cohomology algebra of $SL(n,  \mathbb C)$ is the  exterior algebra on $n-1$ variables $\Lambda (e_3, e_5,...,e_{2n-1})$ if $n\geq 2$ [For $n=1$ your two rings are obviously isomorphic!]. References are given on [this very site][1], as an answer to Evgeny Shinder's question on the cohomology of $GL_n$ and $SL_n$.

**PS** Personally, I much prefer your proof !  But maybe your harsh critic, Mr "some one", will accept the above as satisfying his strange criteria...





[1]:https://mathoverflow.net/questions/18677/cohomology-rings-of-gl-nc-sl-nc