Perhaps the other answer was describing the Picard group of a commutative ring A. If M is a finitely generated projective A-module of rank 1, then the tensor product M ⊗A HomA(M, A) is isomorphic to A. In particular, the set of isomorphism classes of finitely generated projective modules of rank 1 forms a group under tensor product.
If A is an integral domain, then the M we consider are just the isomorphism classes of projective modules that are isomorphic to ideals of A. This forms the ideal class group.
In particular for principal ideal domains or fields, this is not a useful concept.
Also, none of this works for general modules, not even finitely generated projective modules of rank larger than 1, so I would be very cautious of assuming this is true for a negative tensor power of some random module.