On Ext-universal modules in Godel's universe

Let R be an associative unital ring, T a left R-module and lambda an infinite cardinal. We consider the class T-perpendicular to of all left R-modules M satisfying Ext(R)(1) (M , T) = 0 and search for lambda-universal objects in suitable subclasses C of T-perpendicular to. Here, an R-module U is an element of C is lambda-universal for T if vertical bar U vertical bar <= lambda and every R-module M is an element of C of cardinality less than or equal to lambda embeds into U. We show that the existence of vertical bar T vertical bar-universal objects which are strong splitters implies the existence of lambda-universal objects for sufficiently large lambda if we assume (V = L). We then apply our results to module classes over small Dedekind domains to partially solve a generalized problem by Kulikov.