Remark

Tag 05OV, permanent. Cite it as [Bourbaki, Tag 05OV].

Algebra, chapter II, § 3. Tensor products

The tensor product of non-zero modules may be zero: for example, taking the two $\mathbf{Z}$ -modules $E = \mathbf{Z}/2\mathbf{Z}$ and $F = \mathbf{Z}/3\mathbf{Z}$ , $2x = 0$ and $3y = 0$ for all $x \in E$ and $y \in F$ ; therefore, in $E \otimes_{\mathbf{Z}} F$ ,

x \otimes y = 3(x \otimes y) - 2(x \otimes y) = x \otimes (3y) - (2x) \otimes y = 0

for all

x

and

y

(cf. no. 6, Corollary 4 to Proposition 6).

Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.

Cited by

Nothing.

Cites

Proposition 6
no. 6, Corollary 4