Direct product decomposition of semi-hereditary rings (Q2763923)

Let \(R\) be an associative ring with non-zero identity. A submodule \(M\) of a right module \(N\) over the ring \(R\) is called a closed submodule if \(M\) has no proper essential extension in \(N\). Let \(Q\) be an extension ring of the ring \(R\). It is said that the inclusion mapping \(R\to Q\) is a left flat epimorphism if \(Q\) is flat as a right \(R\)-module and the canonical mapping \(Q\otimes_RQ\to Q\) is an isomorphism. The ring \(R\) is said to be a right full linear ring if \(R\) is the ring of all linear transformations of a right vector space over a division ring.NEWLINENEWLINENEWLINEThe main result established by the authors is: Let \(R\) be a right semi-hereditary ring with maximal right quotient ring \(Q\). Then the following conditions are equivalent: (i) Every non-zero closed right ideal contains a minimal closed right ideal and the inclusion mapping \(R\to Q\) is a left flat epimorphism; (ii) \(Q\) is a direct product of right full linear rings and the inclusion mapping \(R\to Q\) is a left flat epimorphism; (iii) \(R\) has a direct product decomposition \(R=\prod_{i\in I}R_i\) such that every \(R_i\) has a maximal right quotient ring \(Q_i\) which is a right full linear ring such that the inclusion mapping \(R_i\to Q_i\) is a left flat epimorphism for every \(i\in I\).