§ Finitely generated as vector space v/s algebra:

As a trivial example, consider K[X]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,X2,}\{ 1, X, X^2, \dots \} is not finite. It is finitely generated as a KK-algebra with generating set {X}\{ X \}.