FUNDAMENTA MATHEMATICAE 185 (2005) Large superdecomposable E(R)-algebras by Laszlo Fuchs (New Orleans) and R¨ udiger G¨ obel (Essen) In honour of Claus Michael Ringel on the occasion of his 60th birthday Abstract. For many domains R (including all Dedekind domains of characteristic 0 that are not fields or complete discrete valuation domains) we construct arbitrarily large superdecomposable R-algebras A that are at the same time E(R)-algebras. Here “superde- composable” means that A admits no (directly) indecomposable R-algebra summands =0 and “E(R)-algebra” refers to the property that every R-endomorphism of the R-module A is multiplication by an element of A. 1. Introduction. Schultz [15] introduced the notion of an E-ring as a ring R such that the endomorphism ring of its additive group is iso- morphic to R under the natural map η → η(1), i.e. each endomorphism acts as multiplication by an element of R. E-rings have been investigated in several papers: see e.g. Dugas–Mader–Vinsonhaler [5], Dugas–G¨ obel [4], G¨ obel–Str¨ ungmann [11], proving the existence of arbitrarily large E-rings, E-rings whose additive groups are ℵ 1 -free abelian groups, etc. G¨ obel–Str¨ ungmann [11] discusses E(R)-algebras, i.e. algebras A over a domain R such that every endomorphism of A as an R-module is multipli- cation by an element of A. The existence of large E(R)-algebras over many domains R is established. Fuchs–Lee [7] constructs E(R)-algebras over cer- tain domains R that are superdecomposable as R-algebras in the sense that they do not admit any algebra summand that is not a direct product of two non-zero subalgebras. In Theorem 5.3 we give a common generalization of these two results by proving the existence of arbitrarily large superdecom- posable E(R)-algebras that are, in addition, ℵ 1 -free in the sense that every countable subset is contained in a free R-submodule. 2000 Mathematics Subject Classification : Primary 13F99, 13C13; Secondary 03E05. Key words and phrases : superdecomposable algebra, E(R)-algebra, ℵ 1 -free, Black Box, trap. This work is supported by the project No. I-706-54.6/2001 of the German-Israeli Foundation for Scientific Research & Development. [71]