An analogue of hilbert's syzygy theorem for the algebra of one-sided inverses of a polynomial algebra

摘要

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: l.gldim(S _n(A))= l.gldim(A) +n. 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.