Embeddings in Artinian rings and Sylvester rank functions

Using Sylvester rank functions it is shown that a right Noetherian algebra, modulo the torsion ideal determined by elements regular modulo its nil radical, is embeddable in a simple Artinian ring. This is used to show that a right Noetherian algebra which is right Krull homogeneous embeds in a simple Artinian ring, and that a right Noetherian affine algebra satisfying a polynomial identity embeds in a simple Artinian ring. The research of the second author was supported in part by NSF Grant DMS-8317737.