Relatively big projective modules and their applications to direct sum decompositions

Countably generated projective modules that are relatively big with respect to a trace ideal were introduced by P. P\v{r}\'ihoda, as an extension of Bass' uniformly big projectives. It has already been proved that there are a number of interesting examples of rings whose countably generated projective modules are always relatively big. In this paper, we increase the list of such examples, showing that it includes all right noetherian rings satisfying a polynomial identity. We also show that countably generated projective modules over locally semiperfect torsion-free algebras over $h$-local domains are always relatively big. This last result applies to endomorphism rings of finitely generated torsion-free modules over $h$-local domains. As a consequence, we can give a complete characterization of those $h$-local domains of Krull dimension $1$ for which every direct summand of a direct sum of copies of a single finitely generated torsion-free module is again a direct sum of finitely generated modules.