# Finitely generated algebra

I am getting the confusion with the definition of algebra. When we say $A$ is a finitely generated $R$- algebra then is that mean $A$ has a ring structure and finitely generated as an $R$-module. Thanks

• Perhaps language problems again...I shall check that and erase my comment. Thanks. – DonAntonio Jan 16 '13 at 18:43
• I thin I must have been thinking of finite-dimensional algebra...oh, well. – DonAntonio Jan 16 '13 at 18:44

This means that apart from $R$-linear combinations of the elements, we can also take all products of the elements, which may well give us a lot more elements.
As an easy example of an algebra (let's say over a field $k$) that is finitely generated as an algebra but not finitely generated as a module over $k$, we can take the polynomial ring $k[x]$ in one indeterminate. As a $k$-module (ie, a vector space), this is infinite dimensional, so not finitely generated. But as a $k$-algebra, it is generated by the element $x$.
• @Tobias: $A$ is an algebra over a field $F$. For finitely many elements $a_1,a_2,\cdots,a_n$ in the algebra, the sub-algebra generated by $a_1,a_2,\cdots,a_n$ we mean, we take $F$-linear combinations of them as well as their products, am I right? – Groups Oct 26 '15 at 9:41
• @Groups Yes, all possible products and $F$-linear combinations (more precisely, we take the smallest subalgebra containing those elements, which will have a slightly different description if we leave out some of the "usual" requirements on the algebra). – Tobias Kildetoft Oct 26 '15 at 9:43