Tensor product of modules: Difference between revisions

Revision as of 05:39, 5 December 2024

Revision as of 21:45, 27 February 2025

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To check that a tensor product <math> M \otimes_R N </math> is nonzero, one can construct an ''R''-bilinear map <math> f:M \times N \rightarrow G </math> to an abelian group <math> G </math> such that {{tmath|1= f(m,n) \neq 0 }}. This works because if {{tmath|1= m \otimes n = 0 }}, then {{tmath|1= f(m,n) = \bar{f}(m \otimes n) = \bar{(f)}(0) = 0 }}.

For example, to see that {{tmath|1= \Z/p\Z \otimes_Zotimes_{\Z} \Z/p\Z }}, is nonzero, take <math> G </math> to be <math> \Z / p\Z </math> and {{tmath|1= (m,n) \mapsto mn }}. This says that the pure tensors <math> m \otimes n \neq 0</math> as long as <math> mn </math> is nonzero in {{tmath|1= \Z / p\Z }}.

=== For equivalent modules ===