\(n\)-extended quasi-Baer rings. (Q1033997)

Let \(R\) be a ring with 1 and \(n\geq 2\) an integer. Then \(R\) is said to be an \(n\)-extended right (principally) quasi-Baer ring if, for any proper (principal) right ideals \(I_1,I_2,\dots,I_n\) of \(R\), the right annihilator of \((I_1I_2\cdots I_n)\) is generated by an idempotent. An \(n\)-extended left (principally) quasi-Baer ring is similarly defined, and an \(n\)-extended (principally) quasi-Baer ring is an \(n\)-extended right and left (principally) quasi-Baer ring. An \(n\)-extended right PP ring is a ring such that the right annihilator of \((x_1x_2\cdots x_n)\) is generated by an idempotent for nonidentity \(x_1,\dots, x_n\) in \(R\). Similarly, an \(n\)-extended left PP ring and an \(n\)-extended PP ring are defined. Then the authors show that if the ring \(R_n\) of the upper triangular matrices of order \(n\) with a same entry along the main diagonal over \(R\) is an \(m\)-extended right PP ring, then so is \(R\). Also, let \(T=\left(\begin{smallmatrix} S&M\\ 0&R\end{smallmatrix}\right)\) be the ring of generalized triangular \(2\times 2\)-matrices where \(R\) and \(S\) are rings and \(M\) an \((S,R)\)-bimodule. Then an equivalent condition is given for \(T\) being an \(n\)-extended right (principally) quasi-Baer ring. Moreover, the endomorphism ring of a finitely generated and projective module over an \(n\)-extended right (principally) quasi-Baer ring \(R\) is an \(n\)-extended right (principally) quasi-Baer ring. Examples are given related to other classes of quasi-Baer rings.