Contraadjusted modules, contramodules, and reduced cotorsion modules

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.