I'm working through some problems when I stumbled across a question asking about conditions for when the polynomial algebra $k[x_1,\ldots,x_n]$ is also a free $A$-module, where $A$ is some $k$-subalgebra of $k[x_1,\ldots,x_n]$. However, I'm blanking on nontrivial examples.
- What are some examples of $A$ for which this occurs?
- Why are these kinds of algebras important, and where can I look to learn more about this concept?