# finite dimensional modules over algebra

Assume that $k$ is a field, $A$ is a $k$-algebra and $M$ is a $A$-module over $k$. In particular $M$ is a $A$-module and a vector space^{} over $k$, thus we may speak about $M$ being finitely generated^{} as $A$-module and finite dimensional as a vector space. These two concepts are related as follows:

Proposition^{}. Assume that $A$ and $M$ are both unital and additionaly $A$ is finite dimensional. Then $M$ is finite dimensional vector space if and only if $M$ is finitely generated $A$-module.

Proof. ,,$\Rightarrow $” Of course if $M$ is finite dimensional, then there exists basis

$$\{{x}_{1},\mathrm{\dots},{x}_{n}\}\subset M.$$ |

Thus every element of $M$ can be (uniquely) expressed in the form

$$\sum _{i=1}^{n}{\lambda}_{i}\cdot {x}_{i}$$ |

which is equal to

$$\sum _{i=1}^{n}({\lambda}_{i}\cdot 1)\cdot {x}_{i}$$ |

since $M$ and $A$ are unital. This completes^{} this implication^{}, because ${\lambda}_{i}\cdot 1\in A$ for all $i$.

,,$\Leftarrow $” Assume that $M$ is finitely generated $A$-module. In particular there is a subset

$$\{{x}_{1},\mathrm{\dots},{x}_{n}\}\subset M$$ |

such that every element of $M$ is of the form

$$\sum _{i=1}^{n}{a}_{i}\cdot {x}_{i}$$ |

with all ${a}_{i}\in A$. Let $m\in M$ be with the decomposition as above. Now $A$ is finite dimensional, so there is a subset

$$\{{y}_{1},\mathrm{\dots},{y}_{t}\}\subset A$$ |

which is a $k$-basis of $A$. In particular for each $i$ we have

$${a}_{i}=\sum _{j=1}^{t}{\lambda}_{ij}\cdot {y}_{j}$$ |

with ${\lambda}_{ij}\in k$. Thus we obtain

$$m=\sum _{i=1}^{n}{a}_{i}\cdot {x}_{i}=\sum _{i=1}^{n}\left(\sum _{j=1}^{t}{\lambda}_{ij}\cdot {y}_{j}\right)\cdot {x}_{i}=$$ |

$$=\sum _{i=1}^{n}\sum _{j=1}^{t}{\lambda}_{ij}\cdot ({y}_{j}\cdot {x}_{i})$$ |

which shows, that all ${y}_{j}\cdot {x}_{i}\in M$ together make a set of generators^{} of $M$ over $k$ (note that ${y}_{j}$ and ${x}_{i}$ are independent on $m$). Since it is finite, then $M$ is finite dimensional and the proof is complete. $\mathrm{\square}$

Title | finite dimensional modules over algebra |
---|---|

Canonical name | FiniteDimensionalModulesOverAlgebra |

Date of creation | 2013-03-22 19:16:35 |

Last modified on | 2013-03-22 19:16:35 |

Owner | joking (16130) |

Last modified by | joking (16130) |

Numerical id | 4 |

Author | joking (16130) |

Entry type | Definition |

Classification | msc 16S99 |

Classification | msc 20C99 |

Classification | msc 13B99 |