Note on Morita equivalence in ring extensions

It seems that Morita invariance is a useful criterion for judging the importance of the classes of ring extensions concerned. Y. Miyashita introduced the notion of Morita equivalence in ring extensions, and he showed that the classes of $G$-Galois extensions and Frobenius extensions are Morita invariant. After that, S. Ikehata showed that the classes of separable extensions, Hirata separable extensions, symmetric extensions, and QF-extensions are Morita invariant. In this paper, we shall prove that the classes of several extensions are Morita invariant. Further, we will give an example of the class of ring extensions which is not Morita invariant.