A finitely generated $k$-algebra which is not a finitely generated $k$-module

In Cox's Ideals, Varieties, and Algorithms, he defined a $k$-algebra is a ring which contains the field $k$ as a subring. Also: A $k$-algebra is finitely generated if it contains finitely many elements such that every element can be expressed as a polynomial (with coefficients in $k$) in these finitely many elements.

Question: According to this definition, $k[x_1, x_2, ..., x_n]$ is a finitely generated $k$-algebra but it is not a finitely generated $k$-module. Am I right? I have this question because being a $k$-algebra is also a $k$-module.

Yes, you're correct.

The polynomials of the form $x_1^{a_1}\cdots x_n^{a_n}$ form a $k$-basis for $k[x_1,\dots,x_n]$, and there are infinitely many of them.

• do you mean "form a $k$-basis for the module"? – mathworker21 Jun 26 '18 at 2:29
• Well, yes, I am viewing it as a module, but I was calling it "the algebra" to say which module I was talking about. The word "basis" without further context refers to a basis of a module (or vector space). I've edited the answer to clarify. – Dave Jun 26 '18 at 2:31