There isn’t much duality in radical theory - Gardner, B.J. (Algebra and Discrete Mathematics, 2007)
The definitions of radical and semi-simple classes are in a natural sense dual to each other. However, statements dual in the same sense to theorems of radical theory tend to be false. Some insights may nevertheless be gained from consideration of duality, and we illustrate this with a link between additive radicals and semi-simple radical classes.