Rank of a module
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
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satisfy the equality
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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 ![]() |
Comments
The rank of a projective module , as defined here, depends on the choice of
.
Rank of a module. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Rank_of_a_module&oldid=48433