An $n$ manifold $M$ is called irreducible if for each embedding of a standard $(n-1)$ sphere $S^{n-1}$ in $M$ there is an embedding of a standard $n$ ball $D^n$ in $M$ such that the image of the boundary $\partial D^n$ coincides with the image of $S^{n-1}$

In case of dimension three it can be proved that each irreducible 3-manifold is also a prime 3-manifold.