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arXiv · 1006.0455

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

V. V. Bavula
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Abstract

An analogue of Hilbert's Syzygy Theorem is proved for the algebra $\mS_n (A)$ of one-sided inverses of the polynomial algebra $A[x_1, ..., x_n]$ over an arbitrary ring $A$: $$ \lgldim (\mS_n(A))= \lgldim (A) +n.$$ The algebra $\mS_n(A)$ 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 $A$: $$ \wdim (\mS_n(A))= \wdim (A) +n.$$

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V. V. Bavula. 2011-04-01. An analogue of Hilbert's Syzygy Theorem for the algebra of one-sided inverses of a polynomial algebra. https://arxiv.org/abs/1006.0455

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