This was asked as part of an earlier question. But since this part did not attract many answers, I am asking it separately.

We consider the homology definition of an orientation for a manifold, as you define fundamental class., ie as some generator of some homology modules, satisfying some compatibility conditions. See for instance the book of Greenberg and Harper. What does it mean to say that a manifold is orientable, over rings other than $\mathbb Z$?

It is nice when the base ring is $\mathbb{Z}/2\mathbb{Z}$; every manifold is orientable here, and has a unique orientation. And thus you can do Poincare duality, etc.. But what on earth does it mean to have $4$ possible orientations for the circle or real line for instance, when you take the base ring for homology to be $\mathbb{Z}/5\mathbb{Z}$?

Maybe it is just a formalism; maybe we do not really have to bother about orientations except the ones given by $+1$ and $-1$ in a ring, and the rest are just matters of additional generators giving some extra vacuous information. But I keep wondering. I hope somebody can clarify.