$n$-Extended Quasi-Baer Rings

A ring with unit is said to be an $n$-extended right (principally) quasi-Baer ring if, for any proper (principal) right ideals $I_1,\dots,I_n$, where $n\ge2$, the right annihilator of the product $I_{1}\dotsb I_{n}$ is generated by an idempotent. A ring with unit is said to be an $n$-extended right (left) PP-ring if the right (left, respectively) annihilator of the product $x_1\dotsb x_n$, where $n\ge2$, is generated by an idempotent for any nonidentity elements $x_{1},\dots,x_{n}$. The behavior of $n$-extended right (principally) quasi-Baer rings and right PP-rings under various constructions and extensions is studied. These classes of rings are closed with respect to direct products and Morita equivalences. Examples illustrating the theory and outlining its frontiers are presented.