If $M$ is a connected, non-compact $n$-manifold, then $H_i(M;R)=0$ for $i\geq n$. For a proof, see Proposition 3.29 in Hatcher's Algebraic Topology book.
So, if you are going to have $H_n(M;R)=R$, $M$ had better be compact.
Bill Kronholm
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