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Classifying E-rings. (English) Zbl 0728.20047

This paper provides a detailed investigation of E-rings. A necessarily commutative ring R with \(1\neq 0\) is an E-ring if any endomorphism of \(R^+\) is scalar multiplication (on the left) by some element in R. The history of E-rings originates from a paper by Beaumont and Pierce in 1961 and the name E-rings and concentrated studies begin with a paper by Schultz in 1975. Many results have been added since to the theory of E- rings. The existence of classes of complicated E-rings of infinite rank made it clear that they are not classifiable. This is different for torsion-free E-rings of finite rank. The results established in this paper, written by two experts of this field, cover a large portion of earlier work from a global algebraic point of view. This paper can be recommended for mathematicians interested in a fast approach to this interesting area. Some of the attraction is based on its links to algebraic number theory, which has stimulated research of E-rings.
I would like to sketch only few of the many results investigated. The classification is concentrated on torsion-free E-rings of finite rank and “classification” refers to quasi-isomorphism. It is important to know that such quasi-isomorphism classes can be detected from prime ideals in the ring of integers of an algebraic number field. “A second key idea... is the inside that to understand a full subring of an algebraic number field K, it is essential to pass to a subring of the Galois closure of K, where the machinery of Galois theory is available.”
Using the Bowshell-Schultz-reduction, the classification of E-rings R (up to quasi-isomorphism) can be reduced to the classification of subrings of algebraic number fields. Moreover, it is assumed that \({\mathbb{Q}}\otimes R\) is essentially the same as the rational hull \({\mathbb{Q}}R\) of R. If \({\bar {\mathbb{Q}}}\) denotes the field of all algebraic numbers, the Galois group \(G=Gal({\bar {\mathbb{Q}}},{\mathbb{Q}})\) is important. The rings in question are in the set \({\mathfrak R}\) of all subrings of \({\bar {\mathbb{Q}}}\) of finite rank. The mentioned reduction argument allows restrictions to \({\mathfrak S}\), the subset of all \(R\in {\mathfrak R}\) which are strongly indecomposable. The action of G on the integral closure S of R gives rise to a Galois theoretic equivalence on \({\mathfrak R}\) with quotient space \(\bar {\mathfrak R}\). This space \(\bar {\mathfrak R}\) can be made into a partially ordered set which carries the information of our E-rings. A mixture of module theoretic arguments of Galois theory on \(\bar {\mathfrak R}\) is made into a beautiful theory and leads to the desired classification of E-rings and more.
Reviewer: R.Göbel (Essen)

MSC:

20K15 Torsion-free groups, finite rank
20K30 Automorphisms, homomorphisms, endomorphisms, etc. for abelian groups
11R04 Algebraic numbers; rings of algebraic integers
16S50 Endomorphism rings; matrix rings
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References:

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