# Skeletons, bodies and generalized $E(R)$-algebras

### Rüdiger Göbel

Universität Duisburg-Essen, Germany### Saharon Shelah

The Hebrew University of Jerusalem, Israel### Daniel Herden

Universität Duisburg-Essen, Germany

## Abstract

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:

*Can we ﬁnd non-commutative generalized E(R)-algebras?*

In Theorem 1.5 we will show that for all countable, principal ideal domains *R* which are not ﬁelds and for any inﬁnite cardinal *κ* there is a non-commutative *R*-algebra *A* of cardinality |*A*| = *κ_ℵ0 with End_R* *A* ≅ *A*, so *A* is a non-commutative generalized *E*(*R*)-algebra, and—not too surprisingly—there is a proper class of examples.

The new strategy should be interesting and useful for other problems as well: We will ﬁrst translate the heart of the algebraic question on the existence of certain monoids via model theory into geometric structures leading to a special class of (decorated) trees and solve this problem introducing products of trees etc. This can be compared with the well-known, but different process of translating group problems to small cancelations in groups via the van Kampen lemma. By small cancelation of trees we are able to ﬁnd a suitable monoid and thus a non-commutative algebra *A* with an important non-canonical embedding *A* → End_R_ *A*, our ∗-scalar multiplication. In a second part of this paper we must enlarge *A* to get rid of all undesired endomorphisms. This can be done more easily (thus ﬁrst) with the help of additional set theory (Jensen’s diamond predictions), which will support the reader to understand more quickly the last steps of the proof. In a ﬁnal chapter we will also give an argument (for removing unwanted endomorphisms) which is based on ordinary set theory of ZFC only (using our favored Black Box predictions; see [20]). Thus we get the result as stated above. Furthermore, this last chapter includes a construction for rigid systems of non-commutative generalized *E*(*R*)-algebras.

## Cite this article

Rüdiger Göbel, Saharon Shelah, Daniel Herden, Skeletons, bodies and generalized $E(R)$-algebras. J. Eur. Math. Soc. 11 (2009), no. 4, pp. 845–901

DOI 10.4171/JEMS/169