# Moscow Mathematical Journal

Volume 17, Issue 3, July–September 2017 pp. 385–455.

Contraadjusted Modules, Contramodules, and Reduced Cotorsion Modules

**Authors**:
Leonid Positselski (1)

**Author institution:**(1) Department of Mathematics, Faculty of Natural Sciences,
University of Haifa, Mount Carmel, Haifa 31905, Israel; and

Laboratory of Algebraic Geometry, National Research
University Higher School of Economics, Moscow 119048; and

Sector of Algebra and Number Theory, Institute for
Information Transmission Problems, Moscow 127051, Russia

**Summary: **

This paper is devoted to the more elementary aspects of
the contramodule story, and can be viewed as an extended introduction
to our more technically complicated paper “Dedualizing complexes and
MGM duality”. Reduced cotorsion abelian groups form an abelian category, which is in some sense covariantly dual to the category of torsion
abelian groups. An abelian group is reduced cotorsion if and only if it
is isomorphic to a product of *p*-contramodule abelian groups over prime
numbers *p*. Any *p*-contraadjusted abelian group is *p*-adically complete,
and any *p*-adically separated and complete group is a *p*-contramodule,
but the converse assertions are not true. In some form, these results hold
for modules over arbitrary commutative rings, while other formulations
are applicable to modules over one-dimensional Noetherian rings.

2010 Math. Subj. Class. 13C12, 13C60, 13D07, 13D99, 13J10.

**Keywords:**Cotorsion modules, contraadjusted modules, contramodules, abelian categories, adic completions, flat covers, cotorsion envelopes, abelian groups, Noetherian commutative rings of Krull dimension 1.

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