(Module of finite type)
Let \(A\) be a commutative ring. An \(A\)-module \(M\) is of finite type, or finitely generated, if there exist \(k \geq 0\) and \(b_1, \ldots, b_k \in M\) such that every \(b \in M\) can be written as \(b = \sum_{i=1}^{k} a_i b_i\) for some \(a_1, \ldots, a_k \in A\); equivalently, if there is a surjective \(A\)-linear map \(A^{k} \to M\).
By the Hilbert basis theorem (T-YYLPH), every ideal \(I \subseteq \OO(\AA^n_k) = k[x_1, \ldots, x_n]\) is an \(\OO(\AA^n_k)\)-module of finite type: \(I = (b_1, \ldots, b_k)\) for some \(b_1, \ldots, b_k \in I\). The polynomial ring \(k[x]\) is a \(k\)-algebra generated by one element but is not a \(k\)-module of finite type, since \(1, x, x^2, \ldots\) are linearly independent; a morphism \(\Spec B \to \Spec A\) is finite (D-MORFIN) when \(B\) is an \(A\)-module of finite type.