Abstract
The notion of central simple algebra over a field has been generalised to the notion of Azumaya algebra over an arbitrary commutative ring R.
The definition is the following
characterisation: an algebra A which is projective of finite type over R
and such that the canonical map A ⊗F A op to End R(A)
is an isomorphism.
One main goal of this note is to show in a constructive setting that this definition is equivalent to a suitable generalisation of another characterisation: an algebra A which becomes a matrix algebra by an algebraic (resp. separable) extension of F.