Is $\mathbb R[x]$ an ideal in $\mathbb R$ ? To me, $\mathbb R[x]$ an ideal. If I took any element from the ring $\mathbb R$ and any element in $\mathbb R[x]$ then the product of those two elements belong to $\mathbb R[x]$
Now about the Hilbert Basis Theorem. The Hilbert Basis Theorem says that every ideal in the ring of polynomials in finitely many variables has a basis.
I am trying to use an example so that I could convince myself. Suppose that I'm interested in the following polynomial:
$3x^2 + 4xy + 2y^2 + 7x - 5y + 12 = 2y^2 + (4x - 5)y + (3x^2 + 7x + 12)$
The LHS is a polynomial in $\ R[x,y]$ and the RHS is a polynomial in $\ (R[x]) [y]$. If I knew that $R$ has the property that every ideal in $R$ is finitely generated, would that then mean that every ideal in $\ R[x,y]$ and $\ (R[x]) [y]$ is finitely generated? Is this the proper interpretation of Hilbert Basis Theorem?