An analogue of Hilbert’s Syzygy Theorem for the algebra of one-sided inverses of a polynomial algebra
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Abstract:
An analogue of Hilbert’s Syzygy Theorem is proved for the algebra $\mathbb {S}_n (A)$ of one-sided inverses of the polynomial algebra $A[x_1, \ldots , x_n]$ over an arbitrary ring $A$: \[ \textrm {l.gldim}(\mathbb {S}_n(A))= \textrm {l.gldim}(A) +n.\] The algebra $\mathbb {S}_n(A)$ 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 $A$: \[ \textrm {w.dim} (\mathbb {S}_n(A))= \textrm {w.dim} (A) +n.\]References
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Additional Information
- V. V. Bavula
- Affiliation: Department of Pure Mathematics, University of Sheffield, Hicks Building, Sheffield S3 7RH, United Kingdom
- MR Author ID: 293812
- Email: v.bavula@sheffield.ac.uk
- Received by editor(s): June 2, 2010
- Received by editor(s) in revised form: April 1, 2011
- Published electronically: February 2, 2012
- Communicated by: Birge Huisgen-Zimmermann
- © Copyright 2012
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 140 (2012), 3323-3331
- MSC (2010): Primary 16E10, 16S85, 16S99
- DOI: https://doi.org/10.1090/S0002-9939-2012-11177-3
- MathSciNet review: 2929003