Throughout this paper A is a finite von Neumann algebra and tr: A−→ C is a finite normal faithful trace. Recall that a von Neumann algebra is finite if and only if it possesses such a trace. Let l2 (A) be the Hilbert space completion of A which is viewed as a pre-Hilbert space by the inner product〈 a, b〉= tr (ab∗). A finitely generated Hilbert A-module V is a Hilbert space V together with a left operation of A by C-linear maps such that λ· 1A acts by scalar multiplication with λ on V for λ∈ C and there exists a unitary A-embedding of V in l2 (A) n=⊕ n i= 1l2 (A). In the sequel A operates always from the left unless explicitly stated differently. A morphism of finitely generated Hilbert A-modules is a bounded A-operator. Let {fin. gen. Hilb. A-mod.} be the category of finitely generated Hilbert A-modules. This category plays an important role in the construction of L2-invariants of finite connected CW-complexes such as L2-Betti numbers and Novikov-Shubin invariants. For a survey on L2-(co) homology we refer for instance to [18],[32],[44]. More information about L2-invariants can be found for instance in [1],[5],[8],[9][13],[14],[17],[19],[24],[25][26],[29],[30],[31],[34],[40],[43]. These constructions of L2-invariants use the rich functional analytic structure. However, it is a consequence of standard facts about von Neumann algebras that they can be interpreted purely algebraically. Namely, we will prove (see Theorem 2.2) if {fin. gen. proj. A-mod.} denotes the category of finitely generated projective A-modules