Jutta Hausen

Centralizer near-rings that are rings

Given an R -module M , the centralizer near-ring ℳ R ( M ) is the set of all functions f: M → M with f(xr)= f(x)r for all x ∈ M and r∈R endowed with point-wise addition and composition of functions as multiplication. In general, ℳ R (M) is not a ring but is a near-ring containing the endomorphism ring E R (M) of M . Necessary and/or sufficient conditions are derived for ℳ R (M) to be a ring. For the case that R is a Dedekind domain, the R -modules M are characterized for which (i) ℳ R (M) is a ring; and (ii) ℳ R (M) = E R (M) . It is shown that over Dedekind domains with finite prime spectrum properties (i) and (ii) are equivalent.

Most Abelian p-groups are determined by the Jacobson radical of their endmorphism rings

A celebrated theorem due to Baer and Kaplansky states that two abelian p-groups are isomorphic if (and only if) their endomorphism rings are isomorphic [B, K]. In fact, every isomorphism between their endomorphism rings is induced by an isomorphism between the groups. In this note we show that an abelian p-group G is already determined by an ideal, K(G), which is contained in the Jacobson radical J(~(G)) of its endomorphism ring o~(G), provided G has an unbounded basic subgroup. Let K (G) denote the set of all torsion elements of J(g(G)). Then K(G) is a two-sided ideal of ~(G) which we shall call the torsion radical of g(G). The torsion radical is zero when G is a divisible or an elementary abelian p-group. Thus, K(G) does not, in general, reveal the structure of G. However, for p-groups which are unbounded modulo their maximal divisible subgroup, we have the following result.

Abelian groups mapping onto their endomorphism rings

An abelian group G is said to be an E-group if it is the additive group of an E-ring. It is known that G is an E-group if and only if there exists a left E(G)-module isomorphism from G to its endomorphism ring E(G). Groups which are isomorphic to the additive group of their endomorphism rings are called weak E-groups. The purpose of this article is to consider the apparently yet weaker condition that there be a homorphism from G onto the additive group of E(G). Groups satisfying this condition are called EE-groups. The properties of EE-groups are studied and it is shown that they are very similar to E-groups. In fact, it is shown that every EE-group of finite torsion-free rank is a weak E-group, and that for various prominent classes of groups the concepts of EE-group and E-group coincide.

Nonsingular modules and -homogeneous maps

A non-singular R-module M is a ray for the class of all nonsingular modules if every R-homogeneous map from M into a non-singular module is additive. Every essential extension of a non-singular locally cyclic module is a ray. We investigate the structure of rays, and determine those semiprime Goldie-rings for which all non-singular modules are rays and those rings for which the only rays are essential extensions of locally cyclic modules.

How automorphism groups reveal Ulm invariants

Abstract For any group Γ and any integer k ⩾ 0 a characteristic subgroup Ψk(Γ) is defined having the following significance. If Γ is the automorphism group of a reduced abelian p-group, p ⩾ 5, whose kth Ulm invariant is fk then Ψk(Γ) ⩽ Ψk+1(Γ) and Ψ k+1 (Γ) Ψ k (Γ) is the full (either general or projective) linear group of degree fk over the prime field of characteristic p).

The maximal normal

Let p be a prime number and let G be an abelian p-group. Let A be the maximal normal p-subgroup of Aut G and C the maximal p-subgroup of its centre. Let t be the torsion radical of ?(G). Then A = (1 + t)(. The result is new for p = 2 and 3, and the proof is new and valid for all primes p.

p

Given a torsion-free abelian group G , a subgroup A of G is said to be a quasi-summand of G if nG ≤ A ⊕ B ≤ G for some subgroup B of G and some positive integer n . If the intersection of any two quasi-summands of G is a quasi-summand, then G is said to have the quasi-summand intersection property. This is a generalisation of the summand intersection property of L. Fuchs. In this note, we give a complete characterisation of the torsion-free abelian groups (in fact, torsion-free modules over torsion-free rings) with the quasi-summand intersection property. It is shown that such a characterisation cannot be given via endomorphism rings alone but must involve the way in which the endomorphism ring acts on the underlying group. For torsion-free groups G of finite rank without proper fully invariant quasi-summands however, the structure of its quasi-endomorphism ring Q E ( G ) suffices: G has the quasi-summand intersection property if and only if the ring Q E ( G ) is simple or else G is strongly indecomposable.

-subgroup of the automorphism group of an abelian

An abelian group A is said to be E-uniserial if the lattice of fully invariant subgroups of A is a chain.