Torsion-freeness and non-singularity over right p.p.-rings.
Let \(R\) be an associative ring with identity, then a right \(R\)-module \(M\) is called non-singular if \(xK\neq 0\) for all non-zero elements \(x\in R\) and all essential right ideals \(K\) of \(R\). A right non-singular ring \(R\) is called right strongly non-singular if every finitely generated non-singular right \(R\)-module can be embedded in a free right \(R\)-module. A module \(M\) is called torsion-free if \(\text{Tor}_1^R(M,R/Rr)=0\) for all elements \(r\in R\). A ring \(R\) is right p.p. if every principal right ideal is projective, and \(R\) is right Utumi if \(R\) is right non-singular and every non-essential right ideal of \(R\) has a non-zero left annihilator. It is shown in the paper that, for a ring \(R\), the classes of torsion-free and non-singular right \(R\)-modules coincide if and only if \(R\) is a right Utumi p.p.-ring with no infinite set of orthogonal idempotents. The authors give several examples and applications of the main result. In particular, a characterization is given for right strongly non-singular rings \(R\) over which the classes of torsion-free and non-singular right \(R\)-modules coincide.
scientific article; zbMATH DE number 2153335
scientific article