Luigi Salce

ALGEBRAIC ENTROPY FOR ABELIAN GROUPS

The theory of endomorphism rings of algebraic structures allows, in a natural way, a systematic approach based on the notion of entropy borrowed from dynamical systems. Here we study the algebraic entropy of the endomorphisms of Abelian groups, introduced in 1965 by Adler, Konheim and McAndrew. The so-called Addition Theorem is proved; this expresses the algebraic entropy of an endomorphism φ of a torsion group as the sum of the algebraic entropies of the restriction to a φ-invariant subgroup and of the endomorphism induced on the quotient group. Particular attention is paid to endomorphisms with zero algebraic entropy as well as to groups all of whose endomorphisms have zero algebraic entropy. The significance of this class arises from the fact that any group not in this class can be shown to have endomorphisms of infinite algebraic entropy, and we also investigate such groups. A uniqueness theorem for the algebraic entropy of endomorphisms of torsion Abelian groups is proved.

Uniserial modules over valuation rings

This is the first in a series of papers deahng with the theory of modules over valuation rings. It has been observed that a variety of concepts and basic results in abelian group theory can be extended, mutatis mutandis, to modules over valuation rings, and several other results, which require drastic modification for valuation rings, can also be dealt with by suitably generalizing ideas of abelian group theory. Our purpose is to develop techniques for modules over valuation rings; actually, these are the simplest kind of commutative non-noetherian rings. Our point of departure is abelian group theory in the local case when the groups are merely modules over Z,, the integers localized at a prime p, which is a discrete, rank one valuation domain. In the process of generalization, the most attractive and frequently used properties are sacrilied, pleasant properties we are so accustomed to in abelian groups are gone; in return, we learn new features and discover new phenomena in the behavior of modules. Several results are known on modules over valuation rings R; see the references [3-lo] or the survey article 121. This paper is devoted to the study of uniseriul R-modules, i.e., those R-modules in which the submodules form a chain. As far as the simplicity of the structure is concerned, these are second only to cyclic modules. They have been investigated by Shores and Lewis [9]; we make use of their results, especially, their description of endomorphism rings. We study various aspects of these modules with special emphasis on their quasiand pure-injectivity, as well as on the existence of pure uniserial submodules in torsion R-modules. In some cases, as expected, more explicit results can be established only under the additional hypothesis that R is almost maximal.

Groups in the class semigroups of valuation domains

It is shown that the isomorphy classes of the ideals of a valuation domain form a Clifford semigroup, and the structure of this semigroup is investigated. The group constituents of this Clifford semigroup are exactly the quotients of totally ordered complete abelian groups, modulo dense subgroups. A characterization of these groups is obtained, and some realization results are proved when the skeleton of the totally ordered group is given.

Adjoint algebraic entropy

Abstract The new notion of adjoint algebraic entropy of endomorphisms of Abelian groups is introduced. Various examples and basic properties are provided. It is proved that the adjoint algebraic entropy of an endomorphism equals the algebraic entropy of the adjoint endomorphism of the Pontryagin dual. As applications, we compute the adjoint algebraic entropy of the shift endomorphisms of direct sums, and we prove the Addition Theorem for the adjoint algebraic entropy of bounded Abelian groups. A dichotomy is established, stating that the adjoint algebraic entropy of any endomorphism can take only values zero or infinity. As a consequence, we obtain the following surprising discontinuity criterion for endomorphisms: every endomorphism of a compact Abelian group, having finite positive algebraic entropy, is discontinuous. This resolves in a strong way an open question from [7] .

A general notion of algebraic entropy and the rank-entropy

Abstract We give a general definition of a subadditive invariant i of Mod(R), where R is any ring, and the related notion of algebraic entropy of endomorphisms of R-modules, with respect to i. We examine the properties of the various entropies that arise in different circumstances. Then we focus on the rank-entropy, namely the entropy arising from the invariant ‘rank’ for Abelian groups. We show that the rank-entropy satisfies the Addition Theorem. We also provide a uniqueness theorem for the rank-entropy.

Length functions, multiplicities and algebraic entropy

Abstract. We consider algebraic entropy defined using a general discrete length function L and will consider the resulting entropy in the setting of -modules. Then entropy will be viewed as a function on modules over the polynomial ring extending L. In this framework we obtain the main results of this paper, namely that under some mild conditions the induced entropy is additive, thus entropy becomes an operator from the length functions on R-modules to length functions on -modules. Furthermore, if one requires that the induced length function satisfies two very natural conditions, then this process is uniquely determined. When R is Noetherian, we will deduce that, in this setting, entropy coincides with the multiplicity symbol as conjectured by the second named author. As an application we show that if R is also commutative, the L-entropy of the right Bernoulli shift on the infinite direct product of a module of finite positive length has value , generalizing a result proved for Abelian groups by A. Giordano Bruno.

Completely positive matrices and positivity of least squares solutions

Abstract A sufficient condition for a doubly nonnegative matrix to be completely positive is given, in terms of the positivity of the least squares solution of a linear system associated to the matrix. Some known results on completely positive matrices are derived by this condition.

Prebasic submodules over valuation rings

SummaryBasic concepts in the theory of modules over valuation rings are introduced. The notion of height is used to define indicators of elements, whose irregularities are investigated. The indicator leads to the new notion of smoothness, a property which does not originate from abelian groups. Invariants generalizing the finite Ulm-Kaplansky invariants of abelian p-groups, as well as the Baer invariants for completely decomposable torsion-free abelian groups, are defined, and several results relating these invariants of a module to those of submodules are proved. All these concepts lead to the notion of prebasic submodules, which seems to be the right analogue of the basic subgroups in abelian groups.

Fully inert subgroups of free Abelian groups

A subgroup