A n-manifold $M$ is called prime if for all the factorizations of $M$ , as a connected sum $M=M_1\#M_2$ , one finds that one of the factors $M_1$ or $M_2$ is the n-sphere $S^n$ .

The importance for 3-manifold is the existence and uniqueness prime decomposition theorem which says: Each 3-manifold can be decomposed as a connected sum of prime manifolds.