Suppose that $A\cong \mathbb{R}[f_1,\dots,f_d]$ is a (commutative) affine $\mathbb{R}$-algebra (with identity); where $f_i$ are polynomials $\mathbb{R}[x_1,\dots,x_N]$. When is $A$ a finite-dimensional $\mathbb{R}$-module?
Some examples are
- Simple $\mathbb{R}$-algebras,
- $\mathbb{R}[x]/[x^n]$ (which is an $n$-dimensional vector space)...
$A$ is finite over $\mathbb R$ if and only if each $f_i$ is integral over $\mathbb R$. This is only the case for $f_i \in \mathbb R$, hence the only example is the trivial one.
Note that your second example isn't one since it is not integral and any algebra of the form you describe is integral.