Nonseparating sphere gives a sphere-circle summand ($M\cong N\#(S^2\times S^1)$)

In an orientable three-manifold, an embedded nonseparating two-sphere has a product neighborhood. Join its two boundary spheres by an arc outside that neighborhood and take a regular neighborhood of the union. This region is $S^2\times S^1$ with an open ball removed. Its boundary sphere exhibits the displayed connected sum. Consequently a prime but nonirreducible orientable three-manifold is $S^2\times S^1$.