A Note on Chain Conditions in Nilpotent Rings and Groups

Maximal and minimal conditions for ideals in associative rings have often been considered, but little seems to be known of these conditions in non-associative rings, or of chain conditions on the non-normal subgroups of a group. Moreover, it is usual to assume the condition for one-sided ideals in noncommutative rings, and the weaker condition for two-sided ideals rarely appears. In this note we first consider a class of groups which are "nilpotent" with respect to a set of operators 0. These groups include ordinary nilpotent groups, and associative and non-associative nilpotent rings and algebras as special cases. Our main theorem is to the effect that, for an Q-nilpotent group, the maximal or minimal condition for O-subgroups implies the corresponding condition for all subgroups. As immediate consequences of this theorem it follows that, for nilpotent rings and algebras, the maximal or minimal condition for ideals implies the corresponding condition for modules, while for nilpotent groups, the maximal or minimal condition for normal subgroups implies the corresponding condition for all subgroups.