Let [itex]M[/itex] be a module over the commutative ring [itex]K[/itex] with unit 1. I want to prove that [itex]M \cong M \otimes K.[/itex] Define [itex]\phi:M \rightarrow M \otimes K[/itex] by [itex]\phi(m)=m \otimes 1.[/itex] This is a morphism because the tensor product is K-linear in the first slot. It is also easy to show that the map is surjective. This is where I get stuck.(adsbygoogle = window.adsbygoogle || []).push({});

Suppose [itex]\phi(m)=\phi(n),[/itex] so that [itex]0 = m \otimes 1 - n \otimes 1 = (m-n) \otimes 1.[/itex] How do I prove that this implies that [itex]m=n[/itex] and thus the map is injective? More generally, how can you tell when a tensor is 0?

**Physics Forums - The Fusion of Science and Community**

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# When is a tensor 0?

Loading...

Similar Threads - tensor | Date |
---|---|

A Tensor symmetries and the symmetric groups | Feb 9, 2018 |

I Tensors vs linear algebra | Jan 28, 2018 |

B Tensor Product, Basis Vectors and Tensor Components | Nov 24, 2017 |

I Matrix for transforming vector components under rotation | Sep 17, 2017 |

Insights What Is a Tensor? - Comments | Jun 18, 2017 |

**Physics Forums - The Fusion of Science and Community**