Test ideal restricted to principal ideals and connections to trace ideals

Motivated by some recent results of F. Pérez and R. R.G connecting test ideal of module closure operations and trace ideals, we investigate the test ideal restricted to principal ideals corresponding to a module closure operation of a torsion-free module $M$. We show that such a test ideal is always a trace ideal, and moreover when $M$ is torsionless and faithful, then it is the trace of the center of ${\mathrm{End}}_R(M)$. Some applications are given.