Assume that k is an algebraically closed field and A is a finite-dimensional wild k-algebra. Recently, L. Gregory and M. Prest proved that in this case the width of the lattice of all pointed A-modules is undefined. Hence the result of M. Ziegler implies that there exists a super-decomposable pure-injective A-module, if the base field k is countable. Here we give a different and straightforward proof of this fact. Namely, we show that there exists a special family of pointed A-modules, called an independent pair of dense chains of pointed A-modules. This also yields the existence of a super-decomposable pure-injective A-module.
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