# Linear Algebra over a Ring/Chain complexes of finitely generated free modules

**Proposition (every chain complex of finitely generated free modules over a Bézout domain is the direct sum of some of its subcomplexes with at most two nonzero terms)**:

Let

be a chain complex whose objects are finitely generated free modules over a Bézout domain . This chain complex is then the countable direct sum of chain complexes of the form

- ,

where and .

**Proof:** We shall construct a direct sum decomposition

- ,

where . Once this is accomplished, we have in fact obtained a direct sum decomposition of the initial chain complex, because elements of are mapped to zero by , and elements of are mapped to due to the chain complex condition .

In order to achieve this decomposition, we invoke Dedekind's theorem for Bézout domains, which tells us that is finitely generated and free; indeed, it is finitely generated, since a generating set is given by the image (via ) of a generating set of . Let thus be a basis of . For each , we choose an arbitrary, but fixed , and then we define . This yields the desired direct sum decomposition. Indeed, , since whenever

for some elements of , applying to both sides of this equation and using its -linearity yields

- ,

which implies . Moreover, , since if is arbitrary, we may select such that

- ,

from which we may easily deduce that

- .

**Proposition (every chain complex of finitely generated free modules over a Bézout domain splits as the direct sum of two types of elementary chain complexes)**:

Let

be a chain complex whose objects are finitely generated free modules over a Bézout domain . This chain complex is the countable direct sum of copies of the following two chain complexes:

- for an element , ie. the arrow represents the function which is given by multiplication by

**Proof:** Using the notation of the last theorem, we have , where is finitely generated and is sent to zero by .