Morita Contexts for Corings and Equivalences

In this note we study Morita contexts and Galois extensions for corings. For a coring $\text{QTR}calC$ over a (not necessarily commutative) ground ring $A$ we give equivalent conditions for $\text{QTR}cal{M}^{\text{QTR}calC}$ to satisfy the weak. resp. the strong structure theorem. We also characterize the so called \QTR{em}{cleft}$C$\QTR{em}{-Galois extensions} over commutative rings. Our approach is similar to that of Y. Doi and A. Masuoka in their work on (cleft) $H$-Galois extensions.