Rank of a module

The rank of a left module $ M $ over a ring $ R $ imbeddable in a skew-field $ k $ is the dimension of the tensor product $ k \otimes _ {R} M $, regarded as a vector space over $ k $. If $ R = \mathbf Z $, the ring of integers, the definition coincides with the usual definition of the rank of an Abelian group (cf. Rank of a group). If $ k $ is a flat $ R $- module (say, $ k $ is the skew-field of fractions of $ R $, cf. Flat module), then the ranks of the modules in an exact sequence

$$ 0 \rightarrow M ^ \prime \rightarrow M \rightarrow M ^ {\prime\prime} \rightarrow 0 $$

satisfy the equality

$$

\mathop{\rm rk}  M =  \mathop{\rm rk}  M  ^  \prime  +  \mathop{\rm rk}  M  ^ {\prime\prime} .

$$

The rank of a free module $ M $ over an arbitrary ring $ R $( 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 $ n $- FI-rings) such that any free module over such a ring with at most $ n $ free generators has a uniquely-defined rank, while for free modules with more than $ n $ generators this property does not hold. A sufficient condition for the rank of a free module over a ring $ R $ to be uniquely defined is the existence of a homomorphism $ \phi : R \rightarrow k $ into a skew-field $ k $. In this case the concept of the rank of a module can be extended to projective modules as follows. The homomorphism $ \phi $ induces a homomorphism of the groups of projective classes $ \phi ^ {*} : K _ {0} R \rightarrow K _ {0} k \approx \mathbf Z $, and the rank of a projective module $ P $ is by definition the image of a representative of $ P $ in $ \mathbf Z $. Such a homomorphism $ \phi $ exists for any commutative ring $ R $.

References

[1]	P.M. Cohn, "Free rings and their relations" , Acad. Press (1971)
[2]	J.W. Milnor, "Introduction to algebraic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077470/r07747029.png" />-theory" , Princeton Univ. Press (1971)

Comments

The rank of a projective module $ P $, as defined here, depends on the choice of $ \phi $.