\(\aleph\)-QF 3 rings with zero singular ideal
A ring R is called right \(\aleph\)-QF 3 if there is a family \(\{e_{\lambda}\}_{\lambda \in \Lambda}\) of nonisomorphic orthogonal idempotents \(e_{\lambda}\) of R with \(Card(\Lambda)=\aleph\) such that each \(e_{\lambda}R\) is the injective envelope of a minimal right ideal and \(\sum e_{\lambda}R\) is faithful. The first main theorem consists of the following equivalent statements: (a) R is right nonsingular and right \(\aleph\)-QF 3; (b) R is right nonsingular, \(E(R_ R)\) is torsionless, and R has essential right socle; (c) R has essential right socle and an R-module is torsionless iff it is nonsingular; (d) R is a two-sided essential product (in the sense of \textit{K. R. Goodearl} [''Ring theory. Nonsingular rings and modules'' (1976; Zbl 0336.16001)]) of right nonsingular right QF 3 rings. The second main theorem consists of several conditions which are equivalent to R being right nonsingular and right and left \(\aleph\)-QF 3. One of these conditions is that R has a two-sided maximal ring of quotients which is a product of simple artinian rings and has essential left and right socles.
scientific article; zbMATH DE number 3975142
scientific article