The lattice of primary ideals of orders in quadratic number fields

Let $O$ be an order in a quadratic number field $K$ with ring of integers $D$, such that the conductor $\mathfrak F = f D$ is a prime ideal of $O$, where $f\in\mathbb Z$ is a prime. We give a complete description of the $\mathfrak F$-primary ideals of $O$. They form a lattice with a particular structure by layers; the first layer, which is the core of the lattice, consists of those $\mathfrak F$-primary ideals not contained in $\mathfrak F^2$. We get three different cases, according to whether the prime number $f$ is split, inert or ramified in $D$.Comment: Keywords: Orders, Conductor, Primary ideal, Lattice of ideals. to appear in Int. J. Number Theory (2016

PRINC domains and comaximal factorization domains

The notion of PRINC domain was introduced by Salce and Zanardo (2014), motivated by the investigation of the products of idempotent matrices with entries in a commutative domain. An integral domain R is a PRINC domain if every two-generated invertible ideal of R is principal. PRINC domains are closely related to the notion of unique comaximal factorization domain, introduced by McAdam and Swan (2004). In this article, we prove that there exist large classes of PRINC domains which are not comaximal factorization domains, using diverse kinds of constructions. We also produce PRINC domains that are neither comaximal factorization domains nor projective-free

Algebraic entropy for valuation domains

AbstractLet R be a non-discrete Archimedean valuation domain, G an R-module, Φ ∈ EndR(G).We compute the algebraic entropy entv(Φ), when Φ is restricted to a cyclic trajectory in G. We derive a special case of the Addition Theorem for entv, that is proved directly, without using the deep results and the difficult techniques of the paper by Salce and Virili [8]

Minimal Pr\"ufer-Dress rings and products of idempotent matrices

We investigate a special class of Pr\"ufer domains, firstly introduced by Dress in 1965. The {\it minimal Dress ring} $D_K$, of a field $K$, is the smallest subring of $K$ that contains every element of the form $1/(1+x^2)$, with $x\in K$. We show that, for some choices of $K$, $D_K$ may be a valuation domain, or, more generally, a B\'ezout domain admitting a weak algorithm. Then we focus on the minimal Dress ring $D$ of $\mathbb{R}(X)$: we describe its elements, we prove that it is a Dedekind domain and we characterize its non-principal ideals. Moreover, we study the products of $2\times 2$ idempotent matrices over $D$, a subject of particular interest for Pr\"ufer non-B\'ezout domains

Fully Inert Subgroups of Abelian p-Groups

A subgroup H of an Abelian group G is said to be fully inert in G, if for every endomorphism ϕ of G, the factor group (H+ϕ(H))/H role= presentation \u3e is finite. This notion arises in the study of the dynamical properties of endomorphisms (entropy). The principal result of this work is that fully inert subgroups of direct sums of cyclic p-groups are commensurable with fully invariant subgroups of the direct sum

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 $\phi$ of a torsion group as the sum of the algebraic entropies of the restriction to a $\phi$-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

Increase in Carotid Intima-Media Thickness in Grade I Hypertensive Subjects

We studied 74 never-treated grade I hypertensive subjects aged 18 to 45 years and 20 normotensive control subjects to define the rate of increase in carotid intima-media thickness (IMT) and the potential role played by the various risk factors. IMT was assessed as mean IMT and as maximum IMT in the right and left common carotid artery, carotid bulb, and internal carotid artery at baseline and at the 5-year follow-up. In grade I hypertensive subjects, both mean IMT and mean of maximum IMT were significantly higher compared with baseline values. Compared with normotensive subjects, both mean IMT and maximum IMT increased significantly (at least P <0.01) in each carotid artery segment. The increase in cumulative IMT was 3.4-fold for mean IMT and 3.2-fold for mean of maximum IMT. Levels of mean arterial pressure at 24-hour monitoring and total serum cholesterol were factors potentially linked to the increment in mean IMT and mean of maximum IMT. Age was also relevant for the increment in mean of maximum IMT, whereas body mass index played some role in the increment of mean IMT. During the follow-up, mean IMT and mean of maximum IMT increased to a greater degree in white-coat hypertensive subjects (n=35) and sustained hypertensive subjects (n=39) than in normotensive control subjects. No differences were found between white-coat hypertensive subjects and sustained hypertensive subjects for both mean IMT and maximum IMT. Levels of mean arterial pressure at 24-hour monitoring affected the increment in IMT in both white-coat hypertensive subjects and sustained hypertensive subjects. In conclusion, our findings indicate that carotid IMT is greater and grows faster in white-coat hypertensive subjects than in normotensive subjects without significant differences with sustained hypertensive patients