An analogue of Hilbert’s Syzygy Theorem for the algebra of one-sided inverses of a polynomial algebra

Abstract

Abstract. An analogue of Hilbert's Syzygy Theorem is proved for the algebra S n (A) of one-sided inverses of the polynomial algebra A[x 1 , . . . , x n ] over an arbitrary ring A:The algebra S n (A) is noncommutative, neither left nor right Noetherian and not a domain. The proof is based on a generalization of the Theorem of Kaplansky (on the projective dimension) obtained in the paper. As a consequence it is proved that for a left or right Noetherian algebra A:w.dim(S n (A)) = w.dim(A) + n.