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Rank of a module

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The rank of a left module over a ring imbeddable in a skew-field is the dimension of the tensor product , regarded as a vector space over . If , the ring of integers, the definition coincides with the usual definition of the rank of an Abelian group (cf. Rank of a group). If is a flat -module (say, is the skew-field of fractions of , cf. Flat module), then the ranks of the modules in an exact sequence

satisfy the equality

The rank of a free module over an arbitrary ring (cf. Free module) is defined as the number of its free generators. For rings that can be imbedded into skew-fields this definition coincides with that in 1). In general, the rank of a free module is not uniquely defined. There are rings (called -FI-rings) such that any free module over such a ring with at most free generators has a uniquely-defined rank, while for free modules with more than generators this property does not hold. A sufficient condition for the rank of a free module over a ring to be uniquely defined is the existence of a homomorphism into a skew-field . In this case the concept of the rank of a module can be extended to projective modules as follows. The homomorphism induces a homomorphism of the groups of projective classes , and the rank of a projective module is by definition the image of a representative of in . Such a homomorphism exists for any commutative ring .

References

[1] P.M. Cohn, "Free rings and their relations" , Acad. Press (1971)
[2] J.W. Milnor, "Introduction to algebraic -theory" , Princeton Univ. Press (1971)


Comments

The rank of a projective module , as defined here, depends on the choice of .

How to Cite This Entry:
Rank of a module. V.E. Govorov (originator), Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Rank_of_a_module&oldid=14427
This text originally appeared in Encyclopedia of Mathematics - ISBN 1402006098