Let \(R\) be a commutative ring with identity and \(M\) an \(n\)-manifold, and for \(A\subseteq M\) write \(H_n(M\mid A;R)\coloneqq H_n(M, M\sm A; R)\), so that \(H_n(M\mid x;R)\cong R\) for \(x\in M\). An \(R\)-orientation of \(M\) is a function \(x\mapsto\mu_x\) assigning to each \(x\in M\) a generator \(\mu_x\) of the \(R\)-module \(H_n(M\mid x;R)\) that is locally consistent: each \(x\in M\) has a neighborhood \(\RR^n\subseteq M\) containing an open ball \(B\) of finite radius about \(x\) and an element \(\mu_B\in H_n(M\mid B;R)\) whose image under the map \(H_n(M\mid B;R)\to H_n(M\mid y;R)\) induced by inclusion is \(\mu_y\) for every \(y\in B\). The manifold \(M\) is \(R\)-orientable if it has an \(R\)-orientation.
Let \(M\) be an \(n\)-manifold.
\(M\) is \(\ZZ/2\)-orientable.
If \(M\) is orientable, then \(M\) is \(R\)-orientable for every commutative ring \(R\) with identity.
If \(R\) is a commutative ring with identity in which \(2\neq 0\), then \(M\) is \(R\)-orientable if and only if \(M\) is orientable.
[@Hat02].