Almost fully decomposable infinite rank lattices over orders

Let Λ be an order over a Dedekind domain R with quotient field K. An object of image, the category of R-projective Λ-modules, is said to be fully decomposable if it admits a decomposition into (finitely generated) Λ-lattices. In a previous article [W. Rump, Large lattices over orders, Proc. London Math. Soc. 91 (2005) 105–128], we give a necessary and sufficient criterion for R-orders Λ in a separable K algebra A with the property that every image is fully decomposable. In the present paper, we assume that image is separable, but that the image-adic completion image is not semisimple for at least one image. We show that there exists an image, such that KL admits a decomposition KL=M0circled plusM1 with image finitely generated, where L∩M1 is fully decomposable, but L itself is not fully decomposable.