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A module over a ring $R$ is pure projective provided it is isomorphic to adirect summand of a direct sum of finitely presented modules. We develop toolsfor the classification of pure projective modules over commutative noetherianrings. In particular, for a fixed finitely presented module $M$, we consider$\mathrm{Add}\, (M)$, which consists of direct summands of direct sums ofcopies of $M$. We are primarily interested in the case where $R$ is aone-dimensional, local domain, and in torsion-free (or Cohen-Macaulay) modules.We show that, even in this case, $\mathrm{Add}\, (M)$ can have an abundance ofmodules that are not direct sums of finitely generated ones.  Our work is based on the fact that such infinitely generated direct summandsare all determined by finitely generated data. Namely, idempotent/trace idealsof the endomorphism ring of $M$ and finitely generated projective modulesmodulo such idempotent ideals. This allows us to extend the classical theorydeveloped to study the behavior of direct sum decomposition of finitelygenerated modules comparing with their completion to the infinitely generatedcase.  We study the structure of the monoid $V^*(M)$, of isomorphism classes ofcountably generated modules in $\mathrm{Add}\, (M)$ with the addition inducedby the direct sum. We show that $V^*(M)$ is a submonoid of $V^*(M\otimes _R\widehat R)$, this allows us to make computations with examples and to provesome realization results.

Big pure projective modules over commutative noetherian rings: comparison with the completion