Definition 2

A commutative algebra $B$ over a commutative ring $A$ is called finitely generated if it is generated by a finite family of elements.

It amounts to the same to say that $B$ is isomorphic to an $A$-algebra of the form $A[X_1, \ldots, X_n]/a$ (where the $X_i$ are indeterminates and $a$ is an ideal of the polynomial ring $A[X_1, \ldots, X_n]$).

If the $A$-algebra $B$ is a finitely generated $A$-module, it is obviously a finitely generated $A$-algebra; the converse is false as the example of polynomial algebras shows (cf. Chapter V).

If $B$ is a finitely generated $A$-algebra and $A'$ is any commutative $A$-algebra, $B_{(A')} = B \otimes_A A'$ is a finitely generated $A'$-algebra, for if $(x_i)_{i \in I}$ is a system of generators of the $A$-algebra $B$, clearly the $x, \otimes 1$ form a system of generators of the $A$-algebra $B_{(A')}$.

If $B$ is a finitely generated $A$-algebra and $C$ a finitely generated $B$-algebra, then $C$ is a finitely generated $A$-algebra; for it follows immediately from the definitions that if $(b_\lambda)_{\lambda \in L}$ is a system of generators of the $A$-algebra $B$ and $(c_\mu)_{\mu \in M}$ is a system of generators of the $B$-algebra $C$, every element of $C$ is equal to a polynomial, with coefficients in $A$, in the $b_\lambda$ and the $c_\mu$.