Remark

Note that in general, if E' is a submodule of a right A-module E and j : E' \to E the canonical injection, the canonical mapping

is not necessarily injective. In other words, for an exact sequence (18) it cannot in general be concluded that the sequence (19) is exact.

Take for example A = \mathbf{Z}, E = \mathbf{Z}, E' = 2\mathbf{Z}, F = \mathbf{Z}/2\mathbf{Z}. As E' is isomorphic to E, E' \otimes F is isomorphic to E \otimes F which is itself isomorphic to F (no. 4, Proposition 4). But for all x' = 2x \in E' (where x \in E) and all y \in F, j(x') \otimes y = (2x) \otimes y = x \otimes (2y) = 0, since 2y = 0, and the canonical image of E' \otimes F in E \otimes F reduces to 0.

In other words, care must be taken to distinguish, for a submodule E' of E and an element x \in E', between the element x \otimes y "calculated in E' \otimes F'" and the element x \otimes y "calculated in E \otimes F" (in other words, the element j(x) \otimes y).

We shall study later, under the name of flat modules, the modules F such that the sequence (19) is exact for every exact sequence (18) (Commutative Algebra, I, § 2).