We generalize Amitsur's construction of central simple algebras over a field F which are split by field extensions possessing a derivation with field of constants F to nonassociative algebras: for every central division algebra D over a field F of characteristic zero there exists an infinite-dimensional unital nonassociative algebra whose right nucleus is D and whose left and middle nucleus are a field extension K of F splitting D, where F is algebraically closed in K. We then give a short direct proof that every p-algebra of degree m, which has a purely inseparable splitting field K of degree m and exponent one, is a differential extension of K and cyclic. We obtain finite-dimensional division algebras over a field F of characteristic p > 0 whose right nucleus is a division p-algebra.
Pumpluen, S. (2018). Algebras whose right nucleus is a central simple algebra. Journal of Pure and Applied Algebra, 222(9), https://doi.org/10.1016/j.jpaa.2017.10.019