Definition 1

In a commutative algebra $\mathbf{B}$ over a commutative ring $\mathbf{A}$, a family $(x_i)_{i \in I}$ of elements of $\mathbf{B}$ is called algebraically free over $\mathbf{A}$ (or the $x_i$ are called algebraically independent over $\mathbf{A}$) if the ideal of algebraic relations between the $x_i$, with coefficients in $\mathbf{A}$, is reduced to 0. A family $(x_i)$ which is not algebraically free over $\mathbf{A}$ is also called algebraically related (or its elements are also called algebraically dependent over $\mathbf{A}$).

This definition generalizes that given in Algebra, Chapter V, § 5, no. 1, Definition 1 for families of elements of a commutative field.

(*) Except in § 5, which uses the results of Chapter I, § 4 and therefore homological algebra, no use is made in this chapter of any books other than Books I to VI and Chapters I and II of this book.

To say that a family $(x_i)_{i \in I}$ is algebraically free over $A$ amounts to the same as saying that the monomials $\prod_i x_i^{n_i}$ in the $x$, are linearly independent over $A$; in particular the $x_i$ are then linearly independent over $A$.