We study the partial ordering on isomorphism classes of central simple algebras over a given field F, defined by setting A1≤A2 if degA1=degA2 and every étale subalgebra of A1 is isomorphic to a subalgebra of A2, and generalizations of this notion to algebras with involution. In particular, we show that this partial ordering is invariant under passing to the completion of the base field with respect to a discrete valuation, and we explore how this partial ordering relates to the exponents of algebras.