Weakly principally quasi-Baer rings. (Q2788747)

Let \(R\) be a ring with \(1\). An idempotent \(e\) is called left (respectively, right) semicentral if \(xe=exe\) (respectively, \(ex=exe\)) for all \(x\in R\). An ideal \(I\) of \(R\) is called right \(s\)-unital by right semicentral idempotents if for every \(x\in I\), \(xe=x\) for some right semicentral idempotent \(e\in I\), and \(R\) is called weakly principally quasi-Baer (= weakly p.q.-Baer) if the left annihilator \(l_R(Ra)\) of \(Ra\) in \(R\) for all \(a\in R\) is right \(s\)-unital by right semicentral idempotents.NEWLINENEWLINE The authors show some properties and characterizations of a weakly p.q.-Baer ring. The following statements are equivalent: (1) \(R\) is weakly p.q.-Baer. (2) For every finitely generated left ideal \(I\) of \(R\), \(l_R(I)\) is right \(s\)-unital by right semicentral idempotents. (3) For every principal ideal \(I\) of \(R\), \(l_R(I)\) is right \(s\)-unital by right semicentral idempotents. (4) For every finitely generated ideal \(I\) of \(R\), \(l_R(I)\) is right \(s\)-unital by right semicentral idempotents. (5) The upper triangular matrix ring of order \(n\) is a weakly p.q.-Baer ring for a positive integer \(n\). (6) \(R[x]\) is a weakly p.q.-Baer ring.NEWLINENEWLINE It is also shown that the weakly p.q.-Baer condition is a Morita invariant property. Moreover, for a prime ideal \(P\) of a weakly p.q.-Baer ring \(R\), let \(O(P)=\{x\in R\mid aRs=0\) for some \(s\notin P\}\) and \(\overline O(P)=\{x\in R\mid x^n\in O(P)\) for some \(n\in N\}\). Then equivalent conditions are given for \(R\) such that every prime ideal contains a unique minimal prime ideal, and when \(O(P)\neq 0\) for every minimal prime ideal \(P\) of \(R\), \(R\) has a nontrivial representation as a subdirect product of the right ring of fractions \(R[S_P^{-1}]\) where \(S_P=\{e\mid e\notin P\) is a left semicentral idempotent\} and \(P\) ranges through all minimal prime ideals.