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Skeletons, bodies and generalized \(E(R)\)-algebras. (English) Zbl 1184.20046

In 1958 in his first book on Abelian groups Fuchs asked the question whether there are rings \(A\) such that \(A\cong\text{End}(A,+)\). If, in particular, the map \(A\to\text{End}(A,+)\): \(a\mapsto\) right multiplication by \(a\) on \(A\) is an isomorphism, then \(A\) is called an \(E\)-ring. It is well known that \(E\)-rings exist even of large cardinalities. \(E\)-rings are necessarily commutative. The authors prove that there are non-commutative rings \(A\) such that \(A\cong\text{End}(A,+)\). These rings cannot be \(E\)-rings and are called quasi-\(E\)-rings.
The authors deal with more general situations and specifically prove the following theorems. Theorem 1.4. Let \(R\) be a cotorsion-free ring with \(1\), and let \(\kappa>|R|\) be a regular, uncountable cardinal with \(\lozenge_\kappa E\) for a non-reflecting stationary subset \(E\subseteq\kappa\) of ordinals cofinal with \(\omega\). Then there is a strongly \(\kappa\)-free, non-commutative \(R\)-algebra \(A\) of cardinality \(|A|=\kappa\) with \(\text{End}_RA\cong A\).
The “Main Theorem” is true in ordinary set theory. Theorem 1.5. Let \(R\) be a cotorsion-free ring with \(1\), and let \(\kappa\geq |R|\) be an uncountable cardinal with \(\kappa=\kappa^{\aleph_0}\). Then there is an \(\aleph_0\)-free, non-commutative \(R\)-algebra \(A\) of cardinality \(|A|=\kappa\) with \(\text{End}_RA\cong A\).
As can be seen from the statements of the theorems there is set theory involved in the proofs. In fact, the proofs are long and involve many concepts such as the skeletons and bodies of the title but also “decorated trees” called types. The algebras are obtained as unions of chains of algebras that are constructed inductively. The proof of the main theorem employs a certain Black Box device to eliminate unwanted endomorphisms.

MSC:

20K30 Automorphisms, homomorphisms, endomorphisms, etc. for abelian groups
20K20 Torsion-free groups, infinite rank
16S50 Endomorphism rings; matrix rings
03E55 Large cardinals
03E75 Applications of set theory
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References:

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