Geometric regularity of direct-sum decompositions in some classes of modules

In this paper, we show that modules with semilocal endomorphism rings appear in abundance in applications, that their direct-sum decompositions are described by the so-called Krull monoids, and that this implies a geometric regularity of the direct-sum decompositions of these modules. Their direct-sum decompositions into indecomposables are not necessarily unique in the sense of the Krull-Schmidt theorem. The application of the theory of Krull monoids to the study of direct-sum decompositions of modules has been developed during the last five years. After a quick survey of the results obtained in this direction, we concentrate in particular on the abundance of examples. At present, these examples are scattered in the literature, and we try to collect them in a systematic way. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 10, No. 3, pp. 231–244, 2004.     

Divisible modules and space of divisibility of an integral domain

Sunto Se A è un modulo divisibile su un dominio d'integrità R e Q è il campo delle frazioni di R, sia R_A Q il massimo sopraanello di R tale che A possa essere dotato della struttura di R_A-modulo. La struttura di R_A riflette quella di A. Se A è un modulo divisibile semplice di torsione, cioè è divisibile di torsione e non ha sottomoduli divisibili propri non nulli, R_A è analiticamente irriducibile. Se C e D sono due moduli divisibili semplici, diciamo C equivalente a D se R_C=R_D. Se d-Space (R) è un insieme di rappresentanti delle classi di equivalenza degli R-moduli divisibili semplici, d-Space (R) puó essere dotato di una struttura di spazio di Hausdorff zero-dimensionale; lo spazio topologico d-Space (R) (da noi chiamato lo spazio di divisibilità di R) contiene un sottospazio chiuso omeomorfo alla superficie di Riemann astratta di R dotata della patch topology.

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Lavoro eseguito con il contribute del Ministero délia Pubblica Istruzione     

Krull-Schmidt fails for serial modules

We answer a question posed by Warfield in 1975: the Krull-Schmidt Theorem does not hold for serial modules, as we show via an example. Nevertheless we prove a weak form of the Krull-Schmidt Theorem for serial modules (Theorem 1.9). And we show that the Grothendieck group of the class of serial modules of finite Goldie dimension over a fixed ring is a free abelian group.

     

Latex Fractionation by Sedimentation and Colloidal Crystallization: The Case of Poly(styrene-co-acrylamide)

Poly(styrene-co-acrylamide) (PS-AAM) latex was prepared, fractionated by sedimentation under gravity, and characterized by PCS, infrared spectra, secondary and backscattered electron imaging in the scanning electron microscope, and electron spectroscopy imaging in an analytical transmission electron microscope. Three latex fractions were obtained. The lower fraction was opalescent and its particles were the more uniform, concerning size, chemical composition, and topochemical features. This lower fraction was still further fractionated by zonal centrifugation in a density gradient, yielding two fractions with similar macrocrystal-forming abilities but different sizes and chemical compositions. These results confirm those previously obtained for the PS-HEMA latex. Copyright 2000 Academic Press.     

Partitioning of metals between the aqueous phase and suspended insoluble material in fog droplets

This paper discusses the partitioning of metals (K, Na, Ca, Mg, Al, Cu, Fe, Pb and Zn) between the aqueous phase and the suspended insoluble material in fog samples collected in the Po Valley during two extensive fields campaigns. Metals represent on average 11% of the mass of suspended insoluble matter, while the main component is carbon (both organic carbon, OC = 35%, and black carbon, BC = 8%). The unaccounted suspended matter mass is very high, on average 46%, and is attributable to non metallic species, such as O and N and of Si. The principal metals in the insoluble suspended fraction are Fe and Al (2-5%), while the contributions of other metals (Na, Mg, Cu, Pb and Zn) are lower than 1%. Ca and K exhibited high blank values and could not be detected above blank detection limit threshold. The main components in the aqueous phase are NO3- (34%), WSOC (23%), SO4(2-) (18%) and NH4+ (19%), while trace metals and remaining cations and anions accounted for less than 1% of solute mass. The main dissolved trace metals in fog droplets are Zn, Al and Fe, while the main metallic cations are Na and Ca. Fe and Al are the only metals preferentially distributed in the suspended insoluble matter of fog droplets (partitioning ratio respectively 37% and 33%). All other metals are mostly dissolved in the aqueous phase (mean partitioning ratios of Mg, Pb, Zn, Cu and Na are 69%, 70%, 77%, 81% and 87%). These findings are in agreement with literature data on metal speciation in cloud and rain samples. The dependence of partitioning ratios on pH is investigated for the different metals, with only Al showing a clear partitioning ratio decrease with increasing pH. Conversely, the other metals show no dependence or a complex and highly variable behaviour. The partitioning ratio of iron (mean 37%) observed in the Po Valley fog samples is much higher than the water extractable iron in aerosol particles (typically 1-2 %): this fact can be explained by differences in the aerosol sources and composition among sites and by chemical processes in the aqueous phase, such as complexation and redox reactions involving organic ligands (oxalate, or other organic acids as humic-like organic matter) which may promote Fe solubility.     

Loewy and Artinian modules over commutative rings

Sunto Si dimostra che se M è un modulo Artiniano con zoccolo semplice su un anello commutativo R, allora End_R(M) è commutativo locale Noetheriano completo e che M, come End_R (M)-modulo, è l'inviluppo iniettivo dell'unico End_R(M)-modulo semplice. Si dimostra anche che ogni modulo di Loewy con invarianti di Loewy finiti è Artiniano e si siudia la successione degli invarianti di Loewy di un tale modulo.