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?
Rank of projective module
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abstract-algebra
modules
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1What properties would you expect such a definition to have? – 2012-04-18
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0For 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
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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