On a generalized Mizuhara construction

We describe the ideals for Mizuhara extensions and find some necessary and sufficient conditions for the simplicity of the direct Mizuhara extension. Also, we study the Mizuhara construction for the matrix algebra and Burde algebras. We construct some various generalizations of the Mizuhara construction and exhibit some examples of the simple pre-Lie algebras that are obtained by this construction; in particular, we construct the simple Witt doubles ${\mathcal A}_d$ and ${\mathcal W}_d({\mathcal A})$ for a unital associative commutative algebra ${\mathcal A}$ with derivation $d$.