Definition 4

— Let M and N be A-modules and assume that M is free. Let Map(M, N) be the A-module of all mappings of M into N. The submodule $\sum_{q > 0} \mathbf{Pol}_A^q(M, N)$ of Map(M, N) is denoted by $\mathbf{Pol}_A(M, N)$ or simply $\mathbf{Pol}(M, N)$; its elements are called polynomial mappings of M into N.

Let $(e_i)_{i \in I}$ be a basis of M and suppose that $l$ is finite; by Prop. 13 (IV, p. 54) a mapping $f$ of M into N is polynomial if and only if there exists a polynomial F in the indeterminates $X_i$ with coefficients in N such that

for every family $x = (x_i)_{i \in I}$ in $A^{(I)}$. This property is independent of the basis chosen for M and it justifies the terminology « polynomial mapping ».