Relatively hereditary radical classes of rings

Radical classes of rings are studied which, while not hereditary, are closed with respect to ideals of some kind: maximal, prime and finite index ideals among others. In some, but not all cases, the ideal property is characterized by the corresponding class of factor rings; for instance maximal ideals are characterized by the simple rings. Such characterizations sometimes make it possible to prove results for several types of ideal simultaneously. Several results concerning hereditary radicals are generalized to various types of relatively hereditary ones, e.g. if <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mstyle mathvariant="script"><mml:mi>R</mml:mi></mml:mstyle></mml:math> is hereditary then for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:mi>I</mml:mi> <mml:mo class="MathClass-bin" stretchy="false">⊲</mml:mo> <mml:mi>A</mml:mi></mml:mrow></mml:math> we have <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:mstyle mathvariant="script"><mml:mi>R</mml:mi></mml:mstyle><mml:mo class="MathClass-open" stretchy="false">(</mml:mo><mml:mi>I</mml:mi><mml:mo class="MathClass-close" stretchy="false">)</mml:mo> <mml:mo class="MathClass-rel" stretchy="false">=</mml:mo> <mml:mi>I</mml:mi> <mml:mo class="MathClass-bin" stretchy="false">∩</mml:mo><mml:mstyle mathvariant="script"><mml:mi>R</mml:mi></mml:mstyle><mml:mo class="MathClass-open" stretchy="false">(</mml:mo><mml:mi>A</mml:mi><mml:mo class="MathClass-close" stretchy="false">)</mml:mo></mml:mrow></mml:math> and hereditary classes define hereditary lower radical classes. In the construction of some examples, use is made of A-radical classes and this leads to some consideration of radical classes of abelian groups.