Prime decompositions of knots in $F \times I$

The famous Schubert Theorem states that any knot in the 3-sphere can be represented as a connected sum of prime knots and the summands are unique. We discuss the question whether a similar theorem is true for knots in thick surfaces. It turns out that the answer is in general negative but positive for homologically trivial knots. We also prove the prime decomposition theorem for virtual knots.