E-groups do not have proper semidirect sum decompositions

A group in which every element commutes with its endomorphic images is called an “E-group″. If p is a prime number, a p-group G which is an E-group is called a “pE-group″. Every abelian group is obviously an E-group. We prove that every 2-generator E-group is abelian and that all 3-generator E-groups are nilpotent of class at most 2. It is also proved that every infinite 3-generator E-group is abelian. We conjecture that every finite 3-generator E-group should be abelian. Moreover, we show that the minimum order of a non-abelian pE-group is p 8 for any odd prime number p and this order is 27 for p = 2. Some of these results are proved for a class wider than the class of E-groups.

In this paper, we study the decompositions of an amply supplemented module M into a direct sum of cotype submodules. For this purpose, we introduce some conatural classes, such as the class of all q-molecular modules, the class of all q-bottomless modules, the class of all q-discrete modules and the class of all q-continuous modules, which are defined for every ring. We also find conditions on the structure of the cotype submodules of an amply supplemented module M, in order for M to have certain direct sum decompositions.

The study of effectiveness in classical mathematics is rapidly expanding, through recent research in algebra, topology, model theory, and functional analysis. Well-known contributors are Barwise (Wisconsin), Crossley (Monash), Dekker (Rutgers), Ershoff (Novosibirsk), Feferman (Stanford), Harrington (Berkeley), Mal′cev (Novosibirsk), Morley (Cornell), Nerode (Cornell), Rabin (Hebrew University), Shore (Cornell). Further interesting work is due to Kalantari (University of California, Santa Barbara), Metakides (Rochester), Millar (Wisconsin), Remmel (University of California, San Diego), Nurtazin (Novosibirsk). Areas investigated include enumerated algebras, models of complete theories, vector spaces, fields, orderings, Hilbert spaces, and boolean algebras.We investigate the effective content of the structure theory of p-groups. Recall that a p-group is a torsion abelian group in which the (finite) order of each element is some power of a fixed prime p. (In the sequel, “group” = “additively written abelian group”.)The structure theory of p-groups is based on the two elementary notions of order and height. Recall that the order of x is the least integer n such that nx = 0. The height of x is the number of times p divides x, that is, the least n such that x = pny for some y in the group but x ≠ pn+1y for any y. If for each n ∈ ω there is a “pnth-root” yn, so that x = pnyn, then we say that x has infinite height. In 1923, Prüfer related the two notions as criteria for direct sum decomposition, provingTheorem. Every group of bounded order is a direct sum of cyclic groups, andTheorem. Every countable primary group with no (nonzero) elements of infinite height is a direct sum of cyclic groups.

This is an extensive synthesis of recent work in the study of endomorphism rings and their modules, bringing together direct sum decompositions of modules, the class number of an algebraic number field, point set topological spaces, and classical noncommutative localization. The main idea behind the book is to study modules G over a ring R via their endomorphism ring EndR(G). The author discusses a wealth of results that classify G and EndR(G) via numerous properties, and in particular results from point set topology are used to provide a complete characterization of the direct sum decomposition properties of G. For graduate students this is a useful introduction, while the more experienced mathematician will discover that the book contains results that are not otherwise available. Each chapter contains a list of exercises and problems for future research, which provide a springboard for students entering modern professional mathematics.

Reading the doctoral thesis of Napp Avelli (2007) I realized that Gabriel's localization theory, which I applied in context with the stabilization of multidimensional input/output behaviors, can also be used for the construction of regular almost interconnections of behaviors in arbitrary dimensions and not only in two dimensions. In this paper I expose this theory in the language of quotient modules and derive an algorithm for arbitrary dimensions which has, however, not yet been implemented. Regular interconnections were introduced and discussed by Willems [IEEE Trans. Automat. Control, 36 (1991), pp. 259–294; 42 (1997), pp. 458–472] for one-dimensional behaviors. Their multidimensional counterparts have been treated by Rocha, Wood, Shankar, Zerz, Lomadze, Napp Avelli, and others since 1998. Two-dimensional almost direct sum decompositions and regular almost interconnections have been considered by Valcher and Bisiacco since 2000 and are also the main subject of Napp Avelli's thesis and his recent submitted paper. Roughly, two behaviors are almost equal if they differ by finite-dimensional behaviors only; so the latter are considered negligible in this context. Two-dimensional behaviors have special properties which are not discussed in the present paper. I also briefly discuss other variants of regular almost interconnections where only stable autonomous behaviors are considered negligible.

The structure of a complex Clifford algebra is studied by direct sum decompositions into eigenspaces of specific linear operators.

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.

A problem of Călugăreanu, Chekhlov and Krylov asks which abelian groups have the property that all of their pure subgroups that are endomorphic images are necessarily summands. Complete answers are given for the groups in certain classes (for example, the torsion groups). On the other hand, examples are constructed that show a complete solution to the general problem is likely to be quite difficult.

The compatible Grassmannian

Direct-Sum Decompositions over Local Rings

The article deals with a Boolean valued approach to some algebraic problems arising from functional analysis. The main results are as follows. (1) A universally complete vector lattice without locally one-dimensional bands contains an infinite direct sum of order dense sublattices each of which is a band preserving linear isomorphic (but not lattice isomorphic) copy of the whole lattice. (2) Every separated injective module over a semiprime rationally complete commutative ring admits a direct sum decomposition with homogeneous summands. (3) A semiprime rationally complete commutative ring properly embedded in a ring with projections K is a homogeneity ring of an additive mapping between appropriate K-modules.

The direct sum behaviour of its projective modules is a fundamental property of any ring. Hereditary Noetherian prime rings are perhaps the only noncommutative Noetherian rings for which this direct sum behaviour (for both finitely and infinitely generated projective modules) is well-understood, yet highly nontrivial . This book surveys material previously available only in the research literature. It provides a re-worked and simplified account, with improved clarity, fresh insights and many original results about finite length modules, injective modules and projective modules. It culminates in the authors' surprisingly complete structure theorem for projective modules which involves two independent additive invariants: genus and Steinitz class. Several applications demonstrate its utility. The theory, extending the well-known module theory of commutative Dedekind domains and of hereditary orders, develops via a detailed study of simple modules. This relies upon the substantial account of idealiser subrings which forms the first part of the book and provides a useful general construction tool for interesting examples. The book assumes some knowledge of noncommutative Noetherian rings, including Goldie's theorem. Beyond that, it is largely self-contained, thanks to the appendix which provides succinct accounts of Artinian serial rings and, for arbitrary rings, results about lifting direct sum decompositions from finite length images of projective modules. The appendix also describes some open problems.

In this paper, the notions of simple, very simple and indecomposable autonomous behaviour are introduced and characterized. By resorting to some recent results about the direct sum decomposition of (linear, time-invariant, differential) behaviours (Bisiacco and Valcher 2001), as well as to the well-known primary decomposition theorem for finitely generated modules (Hartley and Hawkes 1970), it is shown that every autonomous behaviour can be expressed as a direct sum of indecomposable components, which are just cyclic modules of order p v, for some irreducible polynomial p and some positive integer v . The non-uniqueness of this result is also discussed. Finally, this decomposition is interpreted in terms of modal analysis, and related to the results that can be obtained by mean of a state-space realization of the behaviour.

The study of Hopficity in Abelian groups has been largely motivated by the fundamental results of Baumslag, who proved that torsion groups are always Hopfian regardless of their cardinality, but left several questions open concerning torsion-free groups. Later, Corner addressed some of these questions by providing counterexamples showing that a direct sum of two Hopfian groups can be non-Hopfian, and that a group with an automorphism group of order two does not guarantee Hopficity. These results highlighted the need for new constructions to explore Hopficity in torsion-free Abelian groups. Our work introduces a new approach based on divisibility techniques, as it contributes to the understanding of free-torsion groups with respect to the Hopficity property, providing new insights into their structural properties and implications within group theory. Our analysis also demonstrates how divisibility properties, as well as the introduction of totally invariant subgroups and homomorphisms, can be used to establish Hopficity in specific Abelian groups, particularly those that are free-torsion. In order to reach all of this, we start by taking a group defined as an infinite direct sum of cyclic groups; then we construct a specific subgroup generated by two particular families of elements; and finally we show that this group is Hopfian through results from the theory of divisible subgroups.

This article intends to provide a basic study for the complex quaternion ring . We give a precise description of the relations between the norm of the product of two elements in and the product of the norms of the two elements, and develop a new powerful method, namely, the local direct sum decomposition method, to study the ring . Using the “natural basis” introduced by us, we extend the classical rules for matrix calculus to . We also determine the “mixed spherical shell” within which the power series is convergent and without which the power series is divergent.