?-Divisible and ?-flat modules

Ivanov, A. V.

doi:10.1007/bf01140014

Cited by 1 publication

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“…We omit its proof since it uses similar ideas used in the proof of Direct sum of two proper classes are defined in [1]. We say that the sum X In order to show the relation of the classes defined so far with the proper classes related to supplements, we use Ivanov classes and give some results over the ring of integers (see [3,9]). This result is also true for modules over an integral domains since its proof can easily be modified.…”

Section: The Classes O P and O P Rmentioning

confidence: 99%

“…The classes Smal l, S, SB and W S determined by small submodules, submodules that have supplements, submodules that have supplements with bounded intersection and weak supplement submodules respectively, are studied in [4]. Over a hereditary ring, it is shown in the same work that the least proper class W S can be defined using the class W S. In section 3, we give the relation between SB, W S and Ivanov Classes (see [3,9]).…”

Section: Introductionmentioning

confidence: 99%

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Closures of Proper Classes

Alizade¹,

Demirci²

2017

MMN

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For an integral domain R we consider the closures b M. b M r ; r 2 R/ of a submodule M of an R-module N consisting of elements n of N with tn 2 M .r m n 2 M / for some nonzero t 2 R .m 2 Z C / and its connections with usual closure M of M in N. Using these closures we study the closures O P and O P r of a proper class P of short exact sequences and give a decomposition for the class of quasi-splitting short exact sequences of abelian groups into the direct sum of "p-closures" of the class Split of splitting short exact sequences and description of closures of some classes. In the general case of an arbitrary ring we generalize these closures of a proper class P by means of homomorphism classes F and G and prove that under some conditions this closure O P G F is a proper classes.

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Section: The Classes O P and O P Rmentioning

confidence: 99%