This was asked as part of an [earlier question](https://mathoverflow.net/questions/10966/orientability-and-orientation-for-a-differentiable-manifold). 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.



Meaning of orientation/orientability over rings other than the integers
I just wanted to mention that while orientability for cohomology with arbitrary coefficients is governed solely by cohomology with coefficients in ℤ, there are other cohomology theories for which is is not true. For example, if you have an action of $\pi\_1(X)$ on an abelian group M, then you can talk about (co)homology with twisted coefficients in M. For any vector bundle there is a coefficient module such that the bundle is orientable with respect to these twisted coefficients (or, to paraphrase Matthew Ando, "every bundle is orientable if you're twisted enough").


Also, one can ask whether a vector bundle is orientable with respect to topological K-theory, real or complex, or many other generalized cohomology theories, which capture interesting information about the manifold.


So while ℤ/m-coefficients may not be the most interesting coefficient systems to study orientability in, they're part of a larger systematic family of questions (and they don't take much extra work if you're already doing ℤ and ℤ/2-coefficients).


Finally, for something like real coefficients you might think of orientations that differ by a real scalar geometrically, e.g. according to some volume form.


