§ Finitely generated as vector space v/s algebra:

• To be finitely generated as a vector space over $K$ from a generating set $S$ means that we take elements of the form $\sum_i k_i s_i$, or abbreviated, elements of the form $\sum KS$
• To be finitely generated as a $K$ algebra from a generating set $S$, we take elements of the form $\sum_i k_i s_i + \sum_{ij} k_{ij} s_i s_j + \dots$. To abbreviate, elements of the form $\sum C + CS + CS^2 + CS^3 \dots = C/(1-S)$.

As a trivial example, consider $K[X]$. This is not finitely generated as a vector space since it doesn't have a finite basis: the obvious choice of generating set $\{ 1, X, X^2, \dots \}$ is not finite. It is finitely generated as a $K$-algebra with generating set $\{ X \}$.