  Galois closures of commutative rank n ring extensions were introduced by
Bhargava and the second author. In this paper, we generalize the construction
to the case of non-commutative rings. We show that non-commutative Galois
closures commute with base change and satisfy a product formula. As an
application, we give a uniform construction of many of the representations
arising in arithmetic invariant theory, including many Vinberg representations.
