López-Gerena, Juan O.

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Closure operators, torsion theories, and radicals in a non-abelian environment
(2009-05) López-Gerena, Juan O.; Castellini, Gabriele; College of Arts and Sciences - Sciences; Walker Ramos, Uroyoán; Dziobiak, Wieslaw; Department of Mathematics; Castellanos, Dorial
In the category of abelian groups, there exists a relationship between torsion theories and idempotent radicals where we can use one to find the other. In the process we can learn of different properties that apply to the associated subclasses. We can also use a pre-radical to obtain a closure operator and a closure operator to obtain a pre-radical. The properties of the pre-radical (whether it’s idempotent or a radical) indicate corresponding properties of the closure operator and vice-versa. In this thesis, we determine how much of the relationship that exists between closure operators, torsion theories and radicals in the category of abelian groups (Ab) can be extended to the category of all groups (Grp). Once it’s clear which properties do not apply, our goal is to find, for each property, which modifications can be made so that the property still holds.