Geodesic
Skeletons, bodies and generalized $E(R)$-algebras
Journal of the European Mathematical Society, Tome 11 (2009) no. 4, pp. 845-901.

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In this paper we want to solve a fifty year old problem on R-algebras over cotorsionfree commutative rings R with 1. For simplicity (but only for the abstract) we will assume that R is any countable principal ideal domain, but not a field. For example R can be the ring Z or the polynomial ring Q[x]. An R-algebra A is called a generalized E(R)-algebra if its algebra End_R_ A of R-module endomorphisms of the underlying R-module RA is isomorphic to A (as an R-algebra). Properties, including the existence of such algebras are derived in various papers ([5, 6, 9, 10, 20, 22, 24, 25]). The study was stimulated by Fuchs [13], and specially by Schultz [26]. But due to [26] the investigation concentrated on ordinary commutative E(R)-algebras. A substantial part of problem 45 (p. 232) in the monograph [13] (repeated in later publications, e.g. [27]), which will be answered positively for all rings R above in this paper, remained open:
DOI : 10.4171/jems/169
Classification : 20-XX, 16-XX, 00-XX
Keywords: Endomorphism rings, indecomposable modules, E-rings 
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     title = {Skeletons, bodies and generalized $E(R)$-algebras},
     journal = {Journal of the European Mathematical Society},
     pages = {845--901},
     publisher = {mathdoc},
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Rüdiger Göbel; Daniel Herden; Saharon Shelah. Skeletons, bodies and generalized $E(R)$-algebras. Journal of the European Mathematical Society, Tome 11 (2009) no. 4, pp. 845-901. doi : 10.4171/jems/169. http://geodesic.mathdoc.fr/articles/10.4171/jems/169/

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