Tensor product of modules: Difference between revisions

Revision as of 15:46, 13 March 2023

Revision as of 16:01, 13 March 2023

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More category-theoretically, let σ be the given right action of ''R'' on ''M''; i.e., σ(''m'', ''r'') = ''m'' · ''r'' and τ the left action of ''R'' of ''N''. Then, provided the tensor product of abelian groups is already defined, the tensor product of ''M'' and ''N'' over ''R'' can be defined as the [[coequalizer]]:

<math display="block">M \otimes R \otimes N {{{} \atop \overset{\sigma \times 1}\to}\atop{\underset{1 \times \tau} \to \atop {}}} M \otimes N \overset{\otimes}\to M \otimes_R N</math>

where <math>\otimes</math> without a subscript refers to the tensor product of abelian groups.