The classification of the fully invariant subgroups of a reduced Abelian p-group is a difficult long-standing problem when one moves outside of the class of fully transitive groups. In this work we restrict attention to the socles of fully invariant subgroups and introduce a new class of groups which we term socle-regular groups; this class is shown to be large and strictly contains the class of fully transitive groups. The basic properties of such groups are investigated but it is shown that the classification of even this simplified class of groups, seems extremely difficult.

background

Infinite Abelian groups

On the socles of fully invariant subgroups of Abelian p -groups

Let D be a Dedekind domain and R = Int(D) be the ring of integer-valued polynomials of D. We relate the ideal class groups of D and R . In particular we prove that, if D = Z is the ring of rational integers, then the ideal class group of R is a free abelian group on a countably infinite basis. If D is an integral domain with field of fractions K, the ring of integervalued polynomials of D is denoted by Int(D) and is defined to be the subring of K[t] (where t is an indeterminate) consisting of those polynomials f(t) in K[t] such that f(D) C D. Work on rings of integer-valued polynomials has a long history. In particular, Int(7Z) , THE ring of integer-valued polynomials, has been studied at least since the work of Ostrowski [0] and Polya [P]. It was well known even then that Int(Z) is a free module over Z, with a basis consisting of the binomial polynomials Bo (t) , B1 (t), ... , where Bn (t) = t(t 1 ) . .. (t n + 1 )/n! Polya [P] gives a similar result with Z replaced by the ring of algebraic integers in a finite algebraic number field of class number 1. Polya showed that if the integral closure D of Z in a finite algebraic number field is of class number 1 , then Int(D) is a free D-module with a basis consisting of one polynomial of each nonnegative degree. He called such a D-basis for Int(D) a "regular basis". His proof applies for any principal ideal domain D of characteristic zero. Cahen [Ca, ?2] proved that if D is a Dedekind domain, Int(D) is a free D-module, and that if D is a principal ideal domain, then Int(D) has a regular basis. Received by the editors June 26, 1989. This research was presented to the 845th meeting of the AMS, October 28-29, 1988, Lawrence, Kansas, by Professor Heinzer. 1980 Mathematics Subject Classification (1985 Revision). Primary 13B25, 13F05; Secondary 12B05.

https://www.ams.org/proc/1990-108-03/S0002-9939-1990-1009989-7/S0002-9939-1990-1009989-7.pdf

The ring of integer-valued polynomials of a Dedekind domain

On transversal group topologies

The balanced-projective dimension of every abelian p-group is de- termined by means of a structural property that generalizes the third axiom of countability. As a corollary to our general structure theorem, we show for A = Wn that every pA-high subgroup of a p-group G has balanced-projective dimension exactly n whenever G has cardinality Nn but pAG *" O. Our characterization of balanced-projective dimension also leads to new classes of groups where different infinite dimensions are distinguished.

/pdf/the-balanced-projective-dimension-of-abelian-p-groups-lotzqrjqbc.pdf

THE BALANCED-PROJECTIVE DIMENSION OF ABELIAN p-GROUPS

An endomorphisms \({\varphi }\) of an abelian group \(A\) is said inertial if each subgroup \(H\) of \(A\) has finite index in \(H+\varphi (H)\). We study the ring of inertial endomorphisms of an abelian group. We obtain a satisfactory description modulo the ideal of finitary endomorphisms. Also the corresponding problem for vector spaces is considered.

Infinite Abelian groups

Citations

On the socles of fully invariant subgroups of Abelian p -groups

The ring of integer-valued polynomials of a Dedekind domain

On transversal group topologies

THE BALANCED-PROJECTIVE DIMENSION OF ABELIAN p-GROUPS

On the ring of inertial endomorphisms of an abelian group

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