Radical classes, connectednesses and torsion theories

The equality or non-equality of radical classes, torsion classes and connectednesses is investigated in a category that could be any of the concrete categories in which a radical theory, connectednesses theory or a torsion theory have been developed. Among others, it is shown that every connectedness is a radical class. The converse is not true. If, however, a radical class is strict, then it is a connectedness. Furthermore it is proved that every torsion class is a radical class and the converse is not necessarily true. Every radical class with the ADS-property is a torsion class