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On Cleft Extensions of the Tensor Product of Two Hopf Algebras

In this article, we first decompose a cleft extension for X ⊗ Y into two cleft extensions for X and Y respectively, where X and Y are Hopf algebras on a commutative ring R. Conversely, we introduce the concept of consistent cleft Hopf extensions and prove that one can construct a cleft extension for X ⊗ Y by two cleft extensions for X and Y if and only if these two cleft extensions are consistent. An example is also given to show an application of our main results.