7
$\begingroup$

We define the rank of free module as the number of elements on the basis of free module. It may be infinity. How do we define the rank of projective module?

  • 1
    What properties would you expect such a definition to have?2012-04-18
  • 0
    For an arbitrary module $M$ over an *integral domain* $R$, we define the rank of $M$ to be the dimension over $k$ of $k\otimes_RM$, where $k$ is the quotient field of $R$.2012-04-18
  • 1
    @Bruno I would be curious (I haven't thought about it) if we could define the rank $M$ to be the rank of the smallest free module which has $M$ as a direct summand. It seems to work fine for well-behaved rings such as integral domains. Not sure in general. Any ideas?2012-04-18

1 Answers 1