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Right hereditary affine PI rings are left hereditary

Published online by Cambridge University Press:  18 May 2009

Ellen Kirkman  and
James Kuzmanovich
Ellen Kirkman
Affiliation:
Wake Forest University, P.O. Box 7311, Reynolda Station, Winston-Salem, N.C. 27109 U.S.A.
James Kuzmanovich
Affiliation:
Wake Forest University, P.O. Box 7311, Reynolda Station, Winston-Salem, N.C. 27109 U.S.A.
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Small [11] gave the first example of a right hereditary PI ring which is not left hereditary. Robson and Small [9] proved that a prime PI right hereditary ring is a classical order over a Dedekind domain, and hence is Noetherian (and therefore left hereditary). The authors have shown [4] that a right hereditary semiprime PI ring which is finitely generated over its center is left hereditary. In this paper we consider right hereditary PI rings T which are affine (i.e. finitely generated as an algebra over a central subfield k).

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 30 , Issue 1 , January 1988 , pp. 115 - 120
Copyright
Copyright © Glasgow Mathematical Journal Trust 1988

References

1.Albrecht, F., On projective modules over semi-hereditary rings, Proc. Amer. Math. Soc. 12 (1961), 638–639. MR 23 # A3766.Google Scholar
2.Goodearl, K. R., Ring theory: Nonsingular rings and modules, (Marcel Dekker, 1976).Google Scholar
3.Gordon, R. and Small, L. W., Piecewise domains, J. Algebra 23 (1972), 553–564. MR 46 #9087.Google Scholar
4.Kirkman, E. and Kuzmanovich, J., Hereditary finitely generated algebras satisfying a polynomial identity, Proc. Amer. Math. Soc. 83 (1981), 461–466. MR 82k: 16004.Google Scholar
5.Kirkman, E. and Kuzmanovich, J., Matrix subrings having finite global dimension. J. Algebra. 109 (1987), 74–92.Google Scholar
6.Page, A., Sur les anneaux héréditaires ou semi-hereditaires, Comm. Aĺgebra 6 (1978), 1169–1186. MR 57 #12594.Google Scholar
7.Palmér, I. and Roos, J.-E., Formules explicites pour la dimension homologique des anneaux de matrices généralisées, C. R. Acad. Sci. Paris Sér. A–B 273 (1971), A1026–A1029. MR 45 #1977.Google Scholar
8.Procesi, C., Rings with polynomial identities (Marcel Dekker, 1973).Google Scholar
9.Robson, J. C. and Small, L. W., Hereditary prime PI rings are classical hereditary orders, J. London Math. Soc. (2) 8 (1974), 499–503. MR 50 #2236.Google Scholar
10.Sandomierski, F. L., A note on the global dimension of subrings, Proc. Amer. Math. Soc. 23 (1969), 478–480. MR 39 #6930.Google Scholar
11.Small, L. W., An example in Noetherian rings, Proc. Nat. Acad. Sci. U.S.A. 54 (1965), 1035–1036. MR 32 #5691.Google Scholar
12.Small, L. W., Semihereditary rings, Bull. Amer. Math. Soc. 73 (1967), 656–658. MR 35 #2926.Google Scholar