The result indicated by the title
will be proved. More specifically stated: when R is a left order in a simple artinian
ring Q, there exist matrix units {eij} for Q and an element ∕γ ∈ D, where D is the
intersection of the centralizer of {eij} with R, such that rRr ⊆∑Deij and
∑rDeij⊆ R. The Faith-Utumi theorem is an immediate consequence of this
relationship. Furthermore, if R is either a maximal order, or is subdirectly
irreducible, or is hereditary, then there is a left order C in the centralizer of {eij}
which inherits the corresponding property of R and such that R is equivalent to the
matrix ring ∑Ceij.
