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Ira Leviton (talk | contribs) m Deleted a word so that "the" is not repeated and then detected by automated spellcheckers and readers. |
→Construction: This was incorrect. The tensor product of modules is not a quotient of the product of modules, but of the free group generated by the product of modules. Tag: Reverted |
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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 the tensor product of ''M'' and ''N'' over ''R'' can be defined as the [[coequalizer]]:
<math display="block">F(M \times R \times N) {{{} \atop \overset{\sigma \times 1}\to}\atop{\underset{1 \times \tau} \to \atop {}}} F(M \times N) \overset{\otimes}\to M \otimes_R N,</math>
together with the requirements
<math display="block">m \otimes (n + n') = m \otimes n + m \otimes n',</math>
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