Universally torsionless and trace modules

Abstract:

We study, over an arbitrary ring R, a class of right modules intermediate between the projective and the flat content modules. Over the ring of rational integers these modules are the locally free abelian groups. Over any commutative ring they are the modules which remain torsionless under all scalar extensions. They each possess a certain separability property exactly when R is left semihereditary. We define M to be universally torsionless if the natural map $M \otimes A \to {\operatorname {Hom}}({M^\ast },A)$ is monic for all left modules A. We give various equivalent conditions for M to be universally torsionless, one of which is that M is a trace module, i.e. that $x \in M \cdot {M^\ast }(x)$ for all $x \in M$. We show the countably generated such modules are projective. Chase showed that rings over which products of projective or flat modules are also, respectively, projective or flat have other interesting properties and that they are characterized by certain left ideal theoretical conditions. We show similar results hold when the trace or content properties are preserved by products.