KRULL-SCHMIDT THEOREM AND HOMOGENEOUS SEMILOCAL RINGS

Since 2011 in Italy the surface area cultivated with pear tree (Pyrus communis L.) decreased from 35,400 ha to approximately 23,700 ha in 2023, driven by escalating production costs, stagnant market prices, and recurrent phytopathological challenges. Additionally, this decline has been hypothesized to be causally linked to progressively unfavorable climatic conditions, characterized by rising temperatures, frequent summer heatwaves exceeding 35°C, and reduced precipitation. The objective of this study is to identify the most effective soil–plant relationships across Emilia-Romagna, which is the leading Italian region for pear production. We aim at providing a scientific basis for adaptive management strategies linked to regional pedoclimatic variability, so as to support the future of the Italian periculture.Three experimental orchards were located in the provinces of Ferrara, Modena and Ravenna, which are key districts in Italy for periculture were set up in 2023. Agro-meteorological conditions were continuously monitored through an online automated system. Soil samples were thoroughly characterized with respect to their texture, calcium carbonate content, pH, loss on ignition LOI, multi-element profiling by X-ray fluorescence (XRF). To link soil properties with the plant performance, fast chlorophyll a fluorescence was measured in leaves in June 2025 on BA-29 grafted trees and the PItot (total performance index) was calculated, which represents a synthetic indicator of photosystem II efficiency.Soils at the Modena site were predominantly sandy, deriving from Apennine sediments, whereas soils from Ferrara and Ravenna sites exhibited a higher clay content resulting in greater, water-holding capacity. XRF analyses indicate that elemental concentrations at all sites were within expected background levels. LOI and calcimetric analyses were higher in Ravenna soils compared to those from Ferrara and Modena, indicating a greater organic matter and carbonate content. At the Modena site, trees exhibited significantly lower PItot values than those observed in Ferrara and Ravenna sites, This pattern is attributable to the higher sand fraction, which promotes rapid water drainage and reduced nutrient retention in contrast to clay-rich soils where enhanced water and nutrient availability can support improved photosynthetic performance performance. A relationship could also be envisaged found between soil organic carbon content and PItot. Because meteorological conditions were comparable and the germoplasm was uniform across the three sites, the observed differences in photosynthetic performance can be primarily ascribed to soil properties. These location-specific soil–plant correlations can inform precision agriculture practices, rootstock-scion selection, and adaptation strategies to enhance the resilience of pear orchards under changing climate conditions. Research funded by the European Union – NextGenerationEU, Ministero dell’Università e della Ricerca - Piano Nazionale di Ripresa e Resilienza, D.M. 630/2024.

Recreating physiologically relevant spinal cord models in vitro requires not only appropriate cell types but also structural and topographical cues that mimic native tissue. While 2D cultures fail to reproduce the spatial complexity of the central nervous system, 3D bioprinting strategies enable more realistic neural microenvironments. We aim to develop a 3D spinal cord model by combining murine cell lines or human stem cell–derived neural progenitors with a multi-material scaffold system. Murine neuronal/astrocytic (NE-4C) and motor neuron-like (NSC-34) cells were embedded in gelatin methacrylate (GelMA) hydrogels and layered on aligned polycaprolactone (PCL) microfibers produced by Melt Electrowriting via extrusion-based bioprinting. The matrix supported long-term viability, proliferation, and differentiation: neuronal (MAP2, βIII-tubulin) and glial (GFAP) markers increased progressively over 14-21-28 days in vitro (DIV), confirmed by immunofluorescence and western blot, and spontaneous network activity was detected through calcium imaging. In parallel, human neural progenitor cells (NPCs) derived from induced pluripotent stem cells (iPSCs) were successfully differentiated and later cultured within a composite hydrogel composed of GelMA and Geltrex for 7, 10, and 14 DIV. The NPCs expressed SOX2 and nestin, confirming their neural progenitor identity, and exhibited a characteristic rosette- like morphological organization, indicative of early neuroepithelial architecture. Over time, they showed high viability and progressive neuronal maturation, with MAP2, βIIItubulin, and synapsin expression suggesting neuronal interconnection and emerging network formation. This preliminary 3D system might represent a versatile and biomimetic platform for modeling spinal cord structure and function, with translational potential for studies on spinal cord injury, neurodegeneration, and physiologically relevant drug screening. The work was supported by Fondazione CRT and the European Union – Next Generation EU within the PRIN 2022 program (D.D. 104 – 02/02/2022, Ministero dell’Università e della Ricerca).

Abstract. The buildings design is based on a thorough geotechnical study. Knowing the physical and mechanical properties of soils and rocks is a must for engineering calculations. The stability of the building and the lives of people are at risk if the relationship between the structure and the ground is not understood. Geotechnical centrifuges can be used to test physical models for geomeccanics characterization of the land. The first Italian centrifuge - still the only one active in Italy - is that of the former Istituto Sperimentale Modelli e Strutture (ISMES), now Istituto Sperimentale Modelli Geotecnici (ISMGEO). The elastoplastic properties of the soil can be estimated by observing the movement of critical points during the acceleration test. This is possible with traditional transducers, but also thanks to Close Range Multiview Photogrammetry, which allows you to return the digital model of the test before and after the test. By comparing the two models, target displacements can be evaluated, slope surfaces can be measured, and the steepest angle of descent can be determined. Additionally, the study of surface normals enables the identification of the degree of depression and the curved characteristics of the ground, such as anticlinal and synclinal. The contribution focuses on the experiences and outcomes obtained through digital photogrammetry during tests carried out for the SFARS project (Seismic FAilure and post-failure Response of Slopes) funded by the Ministero dell’Istruzione, dell’Università e della Ricerca (MIUR) for the Research Project of National Interest (PRIN 2022).

Strategic Urban Development Under Uncertainty

We study direct product decompositions of objects in a finitely complete and cocomplete category with zero object and certain axioms for a coimage factorization of morphisms. Direct productsC=A×B can be characterized by “inner” properties ofC and its subobjectsA andB. We also show that the Fitting Lemma and the Krull-Schmidt Theorem hold.

This chapter is the beginning of an examination of algebras that are more general than the semisimple algebras. Our strategy is to generalize the process that led to the Wedderburn Structure Theorem. The appropriate substitutes for the basic building blocks of that theory—simple modules—are indecomposable modules. These modules are introduced in this chapter. The analogue of Schur’s lemma provides a characterization of indecomposable modules in terms of their endomorphism algebras. The main result of the chapter is the Krull-Schmidt Theorem. It leads to the conclusion that finitely generated modules over an Artinian algebra decompose uniquely into direct sums of indecomposable modules. In short, the results of Chapter 2 have close analogues in the theory of modules over Artinian algebras.

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.

In this paper, we introduce and study the properties of chain conditions on M-cyclic submodules defined as M-Noetherian and M-Artinian modules and rings. We prove that the property of being iso-M-Noetherian (iso-M-Artinian) is inherited by submodules, quotient modules, and finite sums as well as finite direct sums. Finally, we prove the Hilbert basis theorem for iso-M-Noetherian (iso-M-Artinian) and verify that the property of modules of being M-Noetherian (M-Artinian) and iso-M-Noetherian (iso-M-Artinian) are Morita invariant.

An E-group is a group in which each element commutes with any of its endomorphic images. Hence E-groups provide a generalization of abelian groups. It is easy to show that the endomorphism near-ring (the group generated additively by the endomorphisms) of an E-group is a ring, just as the endomorphism near-ring of an abelian group is a ring. In this paper, we establish the fact that, like abelian groups, E-groups have no proper semidirect sum decompositions (i.e. a semidirect sum decomposition of an E-group must be a direct sum decomposition).

A module [Formula: see text] is called coseparable ([Formula: see text]-coseparable) if for every submodule [Formula: see text] of [Formula: see text] such that [Formula: see text] is finitely generated ([Formula: see text] is simple), there exists a direct summand [Formula: see text] of [Formula: see text] such that [Formula: see text] and [Formula: see text] is finitely generated. In this paper, we show that free modules are coseparable. We also investigate whether or not the ([Formula: see text]-)coseparability is stable under taking submodules, factor modules, direct summands, direct sums and direct products. We show that a finite direct sum of coseparable modules is not, in general, coseparable. But the class of [Formula: see text]-coseparable modules is closed under finite direct sums. Moreover, it is shown that the class of coseparable modules over noetherian rings is closed under finite direct sums. A characterization of coseparable modules over noetherian rings is provided. It is also shown that every lifting (H-supplemented) module is coseparable ([Formula: see text]-coseparable).

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.

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.

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.

Direct-Sum Decompositions over Local Rings