The main purpose of this article is to classify all indecomposable quasi-semiprime comultiplication modules over pullback rings of two Dedekind domains and establish a connection between the quasi-semiprime comultiplication modules and the pure-injective modules over such rings. First, we introduce and study the notion of quasi-semiprime comultiplication modules and classify quasi-semiprime comultiplication modules over local Dedekind domains. Second, we get all indecomposable separated quasi-semiprime comultiplication modules and then, using this list of separated quasi-semiprime comultiplication modules, we show that non-separated indecomposable quasi-semiprime comultiplication Rmodules with finite-dimensional top are factor modules of finite direct sums of separated indecomposable quasi-semiprime comultiplication R-modules.

Keywords: Dedekind domains, Pullback, Pure-injective modules, Separated modules, Quasi-semiprime comultiplication modules.

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