Bimodule structure of central simple algebras

Abstract For a maximal separable subfield K of a central simple algebra A , we provide a semiring isomorphism between K – K -sub-bimodules of A and H – H -sub-bisets of G = Gal ( L / F ) , where F = Cent ( A ) , L is the Galois closure of K / F , and H = Gal ( L / K ) . This leads to a combinatorial interpretation of the growth of dim K ⁡ ( ( K a K ) i ) , for fixed a ∈ A , especially in terms of Kummer subspaces.