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, 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...

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, 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...

Dear Shizuo, 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, 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...

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, 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...