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Cancellation of modules and groups and stable range of endomorphism rings.

The divisor class group of a krull domain

This paper contains a review of results on endomorphism rings of Abelian groups. On one hand, this rapidly developing section of contemporary algebra can be considered as a part of Abelian group theory; on the other hand, it can be considered as a branch of the theory of endomorphism rings of modules. This section is close to both theories, but it has many specific features. There are several important reasons to study endomorphism rings of Abelian groups. First, it provides us with new information on these groups. Second, it stimulates the study of the theory of modules and their endomorphism rings. There are other fields of algebra where the application of endomorphism rings can be useful (additive groups of rings, E-modules and E-rings, and so on). There are already many excellent results in the theory of endomorphism rings of Abelian groups. A large number of methods are used (for example, group, module, categorical, topological, and set-theoretical methods). This survey gives a satisfactory overview of the content and methods of this part of mathematics. One chapter of the book of L. Fuchs [128] is devoted to endomorphism rings of Abelian groups. Also, they are considered in the works of I. Kaplansky [183], A. G. Kurosh [216], D. Arnold [29], and K. Benabdallah [51]. A number of results in this field of algebra are considered in the surveys of A. P. Mishina [244–248], A. V. Mikhalev [242], A. V. Mikhalev and A. P. Mishina [243], V. T. Markov, A. V. Mikhalev, L. A. Skornyakov, and A. A. Tuganbaev [233]. The work of R. Baer [44] has played an important role in the making of the theory of endomorphism rings of Abelian groups and modules. Several fields of ring theory related to endomorphism rings of modules are considered in the works of I. Lambek [218], C. Faith [109, 110], F. Kasch [184], A. A. Tuganbaev [328, 329], and others. However, there does not exist a book which is especially devoted to endomorphism rings; also, there does not exist a systematical presentation of the main results of this theory. This survey slightly fills in this appreciable gap. We note that the book of L. Fuchs does not reflect all the fields of the theory of endomorphism rings. In addition, new sections of this theory appeared after the publication of this book; also, several excellent results were obtained in traditional sections of the theory. In this paper, we consider the main fields of the theory of endomorphism rings of Abelian groups. The most typical results of this theory are included in the review. Some theorems are restated (some statements are presented in more general form, and some theorems are presented in a simplified form). In Secs. 8 and 9, we use the papers of May [235], Gobel [136], and Corner–Gobel [74]. Some parts of these papers are included in the corresponding sections. Our review is designed for specialists in the theory of Abelian groups and the theory of rings and modules. We wish to avoid routine recounting of results and try to give an idea of the possibilities, methods of proofs, and relations between various research fields and individual results. Unsolved problems are presented at the end of every section. Most of them are known; we only systematize them. The authors do not wish to present a complete bibliography; also, there are difficulties related to precedence. Every Abelian group belongs to exactly one of the following three classes of groups: torsion groups, torsion-free groups, and mixed groups. (A group is said to be mixed if it contains a nonzero element of finite order and an element of infinite order.) The properties of endomorphism rings of groups of these three classes are often different. Usually, we emphasize this fact.

Endomorphism Rings of Abelian Groups

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Endomorphism rings and direct sums of torsion free abelian groups

It is proved that if a Lie ring L admits an automorphism of prime order p with a finite number m of fixed points and with pL = L, then L has a nilpotent subring of index bounded in terms of p and m and whose nilpotency class is bounded in terms of p. It is also shown that if a nilpotent periodic group admits an automorphism of prime order p which has a finite number m of fixed points, then it has a nilpotent subgroup of finite index bounded in terms of m and p and whose class is bounded in terms of p (this gives a positive answer to Hartley's Question 8.81b in the Kourovka Notebook). From this and results of Fong, Hartley, and Meixner, modulo the classification of finite simple groups the following corollary is obtained: a locally finite group in which there is a finite centralizer of an element of prime order is almost nilpotent (with the same bounds on the index and nilpotency class of the subgroup). The proof makes use of the Higman-Kreknin-Kostrikin theorem on the boundedness of the nilpotency class of a Lie ring which admits an automorphism of prime order with a single (trivial) fixed point.

Lie rings and lie groups admitting an almost regular automorphism of prime order

A finite rank torsion free abelian group G is almost completely decomposable if there exists a completely decomposable subgroup C with finite index in G The minimum of [G: C] over all completely decomposable subgroups C of G is denoted by i(G) An almost completely decomposable group G has, up to isomorphism, only finitely many summands If i(G) is a prime power, then the rank 1 summands in any decomposition of G as a direct sum of indecomposable groups are uniquely determined If G and H are almost completely decomposable groups, then the following statements are equivalent: (i) G Eb L t H Eb L for some finite rank torsion free abelian group L (ii) i(G) = i(H) and H contains a subgroup G' isomorphic to G such that [H: G ] is finite and prime to i(G) (iii) G ED L % H @ L where L is isomorphic to a completely decomposable subgroup with finite index in G A finite rank torsion free abelian group G is almost completely decomposable if there is a completely decomposable subgroup C having finite index in G It is well known that direct sum decompositions of such groups need not be unique In fact, this class of groups is the source of all the most familiar examples of nonunique decompositions of finite rank torsion free abelian groups This paper will show, however, that the situation is not completely unruly We show that in certain situations cancellation of direct summands is possible We show that a maximal completely decomposable summand is unique up to isomorphism We show ttat an almost completely decomposable group G has, up to isomorphism, only finite many summands We show that there are only finitely many groups H for which there exists a finite rank torsion free abelian group L such that G @ L ; H ( L Theorem 11 characterizes such groups H All groups in this paper, unless indicated otherwise, are finite rank torsion free abelian groups In general, we will follow the notation and convenPresented to the Society, February 2, 1973; received by the editors June 18, 1973 AMS (MOS) subject classifications (1970) Primary 20K15

E. L. Lady

Papers

Endomorphism rings and direct sums of torsion free abelian groups

Almost completely decomposable torsion free abelian groups

Nearly isomorphic torsion free abelian groups

Prime ideals and localization in commutative group rings

Slender rings and modules.