The results in our paper heavily rely on the journal version of [T. Brzeziński, A note on coring extensions, Ann. Univ. Ferrara Sez. VII (N.S.) 51 (2005) 15-27; a corrected version is available at http://arxiv.org/abs/math/0410020v3, Theorem 2.6]. Since it turned out recently that in the proof of the quoted theorem there are some assumptions missing, our derived results are not expected to hold at the stated level of generality either. Here we supplement the constructions in our article with the missing assumptions and show that they hold in most of our examples. In order to handle also the non-fitting case of cleft extensions by arbitrary Hopf algebroids, Morita contexts are constructed that do not necessarily correspond to coring extensions. They are used to prove a Strong Structure Theorem for cleft extensions by arbitrary Hopf algebroids. In this way we obtain in particular a corrected form of the journal version of [G. Böhm, Integral theory for Hopf algebroids, Algebr. Represent. Theory 8 (4) (2005) 563-599; Corrigendum, to be published; see also http://arxiv.org/abs/math/0403195v4, Theorem 4.2], whose original proof contains a similar error.
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