Embeddings in Artinian rings and Sylvester rank functions

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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.