On products of idempotents and nilpotents

This article studies the ring structure arising from products of idempotents and nilpotents. Thus the argument is concerned essentially with the one-sided IQNN property of rings. We first prove that if the $2$ by $2$ full matrix ring over a principal ideal domain F of characteristic zero is right IQNN then F contains infinitely many non-integer rational numbers; and that the concepts of right IQNN and right quasi-Abelian are independent of each other. We next introduce a ring property, called { right IAN}, as a generalization of both right IQNN and right quasi-Abelian; and provide several kinds of methods to construct right IAN rings. In the procedure, we also show that the right IQNN and right IAN do not go up to polynomial rings.