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Mat. Sb. (N.S.), 1975, Volume 96(138), Number 3, Pages 414–446 (Mi msb3398)  

This article is cited in 3 scientific papers (total in 3 papers)

Inductive purities in Abelian groups

A. A. Manovtsev


Abstract: In the paper we study purities $\omega$ in categories of Abelian groups having the property that the union of an increasing chain of $\omega$-pure subgroups of an Abelian group $G$ is itself an $\omega$-pure subgroup of $G$. Such purities are called inductive. For every prime number $p$ we set $A\subseteq_{\eta_p}B$ if for $A\ni a=p^kb$, $b\in B$, there is an $a'\in A$ and an $l\geqslant0$ such that $p^la=p^{k+l}a'$. Head purities are defined as purities of the form $\eta_\Pi=\bigcap_{p\in\Pi}\eta_p$, where $\Pi$ is a set of prime numbers. Head purities and $\varepsilon$-purities, evidently, are inductive. In the paper we show that every inductive purity in the category of all torsion-free Abelian groups is a certain $\Pi$-servancy, every inductive purity in the category of all periodic Abelian groups is a certain $\varepsilon$-purity, and every inductive purity in the category of all Abelian groups is the intersection of a certain $\varepsilon$-purity and a certain Head purity.
Bibliography: 8 titles.

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English version:
Mathematics of the USSR-Sbornik, 1975, 25:3, 389–418

Bibliographic databases:

UDC: 519.443
MSC: 20K99
Received: 25.04.1974

Citation: A. A. Manovtsev, “Inductive purities in Abelian groups”, Mat. Sb. (N.S.), 96(138):3 (1975), 414–446; Math. USSR-Sb., 25:3 (1975), 389–418

Citation in format AMSBIB
\Bibitem{Man75}
\by A.~A.~Manovtsev
\paper Inductive purities in Abelian groups
\jour Mat. Sb. (N.S.)
\yr 1975
\vol 96(138)
\issue 3
\pages 414--446
\mathnet{http://mi.mathnet.ru/msb3398}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=419642}
\zmath{https://zbmath.org/?q=an:0335.20027}
\transl
\jour Math. USSR-Sb.
\yr 1975
\vol 25
\issue 3
\pages 389--418
\crossref{https://doi.org/10.1070/SM1975v025n03ABEH002214}


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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. E. G. Sklyarenko, “Relative homological algebra in categories of modules”, Russian Math. Surveys, 33:3 (1978), 97–137  mathnet  crossref  mathscinet  zmath
    2. A. I. Generalov, “On an axiomatic description of weak purities in the category of modules”, Math. USSR-Sb., 37:1 (1980), 71–82  mathnet  crossref  mathscinet  zmath  isi
    3. Fomin A., “Abelian Groups in Russia”, Rocky Mt. J. Math., 32:4 (2002), 1161–1180  crossref  mathscinet  zmath  isi
  • Математический сборник (новая серия) - 1964–1988 Sbornik: Mathematics (from 1967)
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