Bounded topologies on endomorphism rings

Let R be an associative ring with a unit and N be a left R-module. The set MR(N) = {f: N → N | f(rx) = rf(x), r ∈ R, x ∈ N} is a near-ring with respect to the operations of addition and composition and contains the ring ER(N) of all endomorphisms of the R-module N. The R-module N is endomorphic if MR(N) = ER(N). We call an Abelian group endomorphic if it is an endomorphic module over its endomorphism ring. In this paper, we find endomorphic Abelian groups in the classes of all separable torsion-free groups, torsion groups, almost completely decomposable torsion-free groups, and indecomposable torsion-free groups of rank 2.

It was shown in Huber [4; Theorem 3.3] that an R-module E is slender as a module over its endomorphism ring S = EndR E if it is of the form E = M(IN) for some R-module M. This had important consequences for questions of E-reflexivity. In Mader [6; Lemma 5.1] it was shown that the abelian group E = M ∞i=1 Z(pi)is slender over S = End E and this was a key step in proving that the completion of a group A of non-measurable cardinality equipped with the direct-sum-of-cyclics topology of D’Este [1] is algebraically the group HomS(HomZ (A, E), E). At the Honolulu Conference on Abelian Group Theory 1982/83, Martin Huber suggested to me that [6; Lemma 5.1] should be extendable to more general direct sums. In this note we prove such a theorem and use it to characterize completely the totally projective p-groups which are slender as modules over their endomorphism rings.

The projective dimension of an abelian group with torsion reduced part, as a module over its endomorphism ring, is determined.In particular, a group of projective dimension 2 is exhibited.Any abelian group G may be considered as a module over its endomorphism ring E. It then is natural to study the properties of G as an P-module.In particular, you can try to compute the various homological dimensions of G.For example, in [6] we showed that a /J-group G is flat over E (i.e. has weak dimension zero) unless its reduced part is bounded and its divisible part is nonzero.In [7] we characterized those G which are injective over P. Douglas and Farahat [2] showed that if G is torsion, then dimEG < 1, where dim£G denotes the projective dimension of G over P. Later [3] they

The construction of torsion-free abelian groups with prescribed endomorphism rings starting with Corner’s seminal work (see Corner, 1963) is a well-studied subject in the theory of abelian groups. Usually these construction work by adding elements from a (topological) completion in order to get rid of (kill) unwanted homomorphisms. The critical part is to actually prove that every unwanted homomorphism can be killed by adding a suitable element. We will demonstrate that some of those constructions can be significantly simplified by choosing the elements at random. As a result, the endomorphism ring will be almost surely prescribed, i.e., with probability one.

In a recent paper2 we have called a ring 2 primitive if 2 contains a maximal right ideal a such that the quotient (a: W) = (0). The quotient (: 1) is defined to be the largest two-sided ideal of 21 that satisfies the condition 21(a: 21) C a. Thus if 21lias an identity, (a: 21) is the largest two-sided ideal of 21 contained in a. Primitive rings appear to play a role in the general structure theory of rings analogous to that played by simple rings in the Wedderburn-Artin theory. Primitive rings are also fundamental in general representation theory since it is known that a necessary and sufficient condition that a ring 21 be primitive is that 21 be isomorphic to an irreducible ring W of endomorphisms in a commutative group T. If 2 is irreducible in T and Zi is the division ring of endomorphisms commutative with the elements of A, then R is a dense ring of linear transformations in T regarded as a vector space over Z.3 We shall call 2 a left primitive ring if 2 contains a maximal left ideal a' such that the quotient (}': 21)l = (0). Here (s': 2) 1 is the largest two-sided ideal in 2 having the property (a': 21)1 21 C a'. Evidently, a ring is left primitive if and only if it is anti-isomorphic to an irreducible ring of endomorphisms. It is an open question whether or not a primitive ring is necessarily left primitive. This is indeed the case if 2 contains minimal ideals. In many other respects the theory of primitive rings that contain minimal ideals constitutes the most satisfactory part of the theory of primitive rings. An appropriate tool for studying rings of this type is a generalization of the duality theory previously used by Dieudonn6 in studying simple rings that possess minimal ideals.4 In this note we develop this generalization of Dieudonn6's results. We also investigate certain natural topologies that can be defined in any primitive ring.

Properties of abelian groups related to a given finite rank torsion free abelian group A are analyzed in terms of End (A), the endomorphism ring of A. This point of view gives rise to generalizations of some classical theorems by R. Baer and examples of pathological direct sum decompositions of finite rank torsion free abelian groups.

Introduction. In [7] and [8] we have determined the isomorphisms of the endomorphism rings of two special classes of torsion-free modules. In [7] the modules were free, over principal left ideal domains, and the isomorpnism was always induced by a semi-linear transformation of the modules. In [8] the ring of scalars was a complete discrete valuation ring, but no further restriction was placed on the modules. Except for the special case when both modules are divisible, the same result holds as in [7]. In each case a key point in the argument is that a certain indecomposable direct summand can be shown to be cyclic. Even for ordinary torsion-free abelian groups, the abundance of the indecomposable ones makes it impossible to prove that isomorphism of the endomorphism rings implies isomorphism of the underlying groups. We show first that for any cardinal number c, there exist homogeneous completely decomposable groups G and H of rank c which are not isomorphic, but have isomorphic rings of endomorphisms. This makes it clear that strong restrictions must be placed on the class of groups considered if it is hoped that the ring isomorphism is to be induced by a group isomorphism. In particular, homogeneity coupled with separability or complete decomposability is not sufficient. We are, however, able to generalize the results of [7] to a class of what we call locally free modules A over principal left ideal domains F. A locally free abelian group (F is the ring of integers) is just a homogeneous separable group of null type [6, p. 208]. This latter class of groups includes the complete direct sum of infinite cyclic groups, as well as all its subgroups [4, p. 176]. For example, the set of all bounded sequences of integers is such a group. We also determine (Theorem B) which semi-linear transformations induce the same ring isomorphisms. The results are similar to the vector space case considered in [2], and apply to the endomorphism rings of locally free modules as well as the torsion-free modules of [8].

We use a variation on a construction due to Corner 1965 to construct (Abelian) groups A that are torsion as modules over the ring End (A) of group endomorphisms of A. Some applications include the failure of the Baer-Kaplansky Theorem for Z[X]. There is a countable reduced torsion-free group A such that IA = A for each maximal ideal I in the countable commutative Noetherian integral domain, End (A). Also, there is a countable integral domain R and a countable. R-module A such that (1) R = End(A), (2) T0 ⊗RA ≠ 0 for each nonzero finitely generated (respectively finitely presented) R-module T0, but (3) T ⊗RA = 0 for some nonzero (respectively nonzero finitely generated). R-module T.

Абелевы группы с π-регулярным центром кольца эндоморфизмов

The results of Szele and Szendrei [‘On Abelian groups with commutative endomorphism rings’, Acta Math. Acad. Sci. Hungar.2 (1951), 309–324] characterizing abelian groups with commutative endomorphism rings are generalized to modules whose endomorphism rings have various restrictions on their idempotents. Such properties include central or commuting idempotents, and one-sided ideals being two-sided. Related properties include direct summands having unique complements, or being fully invariant.

If E is a ring, and G is an E-module, then G is quasi-injective if every homomorphism from a submodule of G to G can be extended to an endomorphism of G. POOLE and REID [2] raise the question as to which abelian groups G are quasiinjective as modules over their endomorphism rings E. They prove that direct sums of cyclic p-groups are quasi-injective. This result is extended here to a large class of p-groups, including all p-groups that have no elements of infinite height, and all totally projective p-groups -- in particular, all countable p-groups. The central idea is a property shared by totally projective p-groups and pgroups with no elements of infinite height. DEFINITION. If G is a p-group and xEG[p], then x is said to be detachable if G can be written as G1 @ Gz with (x) =p'G~ [p] for some ordinal c~. If every element of G[p] is detachable, we say that G is detachable. We write p'G~[p] instead of p~G~ to allow Ga to be divisible. Note that every element in G[p] of finite height is detachable, so every p-group with no elements of infinite height is detachable. Hill's extension of Ulm's theorem to totally projective p-groups (see [3]) implies that these groups are also detachable. THEOREM 1. If G is a detachable p-group, and x, yEG[p], then there is an endomorphism f of G such that f(x) =y or f(y) = x.

Endoregular modules

Some years ago A. L. S. Corner proved that every countable and cotorsion-free ring can be realized as the endomorphism ring of some torsion-free abelian group. This result has many interesting consequences for abelian groups. Using a set-theoretic axiom ∇k., which follows for instance from V = L, we can drop the countability condition in Corner's theorem.

In this paper, we determine the locally compact abelian (LCA) groups whose ring of continuous endomorphisms is local.

Given an abelian algebraic group A over a global field F, α ∈ A(F), and a prime ℓ, the set of all preimages of α under some iterate of [ℓ] generates an extension of F that contains all ℓ-power torsion points as well as a Kummer-type extension. We analyze the Galois group of this extension, and for several classes of A we give a simple characterization of when the Galois group is as large as possible up to constraints imposed by the endomorphism ring or the Weil pairing. This Galois group encodes information about the density of primes p in the ring of integers of F such that the order of (α mod p) is prime to ℓ. We compute this density in the general case for several classes of A, including elliptic curves and one-dimensional tori. For example, if F is a number field, A/F is an elliptic curve with surjective 2-adic representation and α ∈ A(F) with α ∉ 2A(F(A[4])), then the density of p with (α mod p) having odd order is 11/21.