Maximal orders and reflexive modules
HTML articles powered by AMS MathViewer
- by J. H. Cozzens PDF
- Trans. Amer. Math. Soc. 219 (1976), 323-336 Request permission
Abstract:
If R is a maximal two-sided order in a semisimple ring and ${M_R}$ is a finite dimensional torsionless faithful R-module, we show that $m = {\text {End}_R}\;{M^\ast }$ is a maximal order. As a consequence, we obtain the equivalence of the following when ${M_R}$ is a generator: 1. M is R-reflexive. 2. $k = {\text {End}}\;{M_R}$ is a maximal order. 3. $k = {\text {End}_R}\;{M^\ast }$ where ${M^\ast } = {\hom _R}(M,R)$. When R is a prime maximal right order, we show that the endomorphism ring of any finite dimensional, reflexive module is a maximal order. We then show by example that R being a maximal order is not a property preserved by k. However, we show that $k = {\text {End}}\;{M_R}$ is a maximal order whenever ${M_R}$ is a maximal uniform right ideal of R, thereby sharpening Faith’s representation theorem for maximal two-sided orders. In the final section, we show by example that even if $R = {\text {End}_k}V$ is a simple pli (pri)-domain, k can have any prescribed right global dimension $\geqslant 1$, can be right but not left Noetherian or neither right nor left Noetherian.References
- Maurice Auslander and Oscar Goldman, Maximal orders, Trans. Amer. Math. Soc. 97 (1960), 1–24. MR 117252, DOI 10.1090/S0002-9947-1960-0117252-7
- Hyman Bass, Finitistic dimension and a homological generalization of semi-primary rings, Trans. Amer. Math. Soc. 95 (1960), 466–488. MR 157984, DOI 10.1090/S0002-9947-1960-0157984-8
- Victor P. Camillo and J. Cozzens, A theorem on Noetherian hereditary rings, Pacific J. Math. 45 (1973), 35–41. MR 318198, DOI 10.2140/pjm.1973.45.35
- Henri Cartan and Samuel Eilenberg, Homological algebra, Princeton University Press, Princeton, N. J., 1956. MR 0077480
- P. M. Cohn, Quadratic extensions of skew fields, Proc. London Math. Soc. (3) 11 (1961), 531–556. MR 136633, DOI 10.1112/plms/s3-11.1.531
- John H. Cozzens, Homological properties of the ring of differential polynomials, Bull. Amer. Math. Soc. 76 (1970), 75–79. MR 258886, DOI 10.1090/S0002-9904-1970-12370-9
- Carl Faith, Noetherian simple rings, Bull. Amer. Math. Soc. 70 (1964), 730–731. MR 167500, DOI 10.1090/S0002-9904-1964-11193-9
- Carl Faith, A correspondence theorem for projective modules and the structure of simple Noetherian rings, Symposia Mathematica, Vol. VIII (Convegno sulle Algebre Associative, INDAM, Rome, 1970) Academic Press, London, 1972, pp. 309–345. MR 0340333 —, Algebra: Rings, modules and categories, Springer-Verlag, New York and Berlin, 1973.
- K. R. Goodearl, Subrings of idealizer rings, J. Algebra 33 (1975), 405–429. MR 357510, DOI 10.1016/0021-8693(75)90110-6
- R. Hart and J. C. Robson, Simple rings and rings Morita equivalent to Ore domains, Proc. London Math. Soc. (3) 21 (1970), 232–242. MR 279131, DOI 10.1112/plms/s3-21.2.232
- Nathan Jacobson, The Theory of Rings, American Mathematical Society Mathematical Surveys, Vol. II, American Mathematical Society, New York, 1943. MR 0008601, DOI 10.1090/surv/002
- James P. Jans, Rings and homology, Holt, Rinehart and Winston, New York, 1964. MR 0163944
- Arun Vinayak Jategaonkar, Endomorphism rings of torsionless modules, Trans. Amer. Math. Soc. 161 (1971), 457–466. MR 284464, DOI 10.1090/S0002-9947-1971-0284464-9
- Eben Matlis, Reflexive domains, J. Algebra 8 (1968), 1–33. MR 220713, DOI 10.1016/0021-8693(68)90031-8
- Bruno J. Müller, The quotient category of a Morita context, J. Algebra 28 (1974), 389–407. MR 447336, DOI 10.1016/0021-8693(74)90048-9
- J. C. Robson, Idealizers and hereditary Noetherian prime rings, J. Algebra 22 (1972), 45–81. MR 299639, DOI 10.1016/0021-8693(72)90104-4
- Julius Martin Zelmanowitz, Endomorphism rings of torsionless modules, J. Algebra 5 (1967), 325–341. MR 202766, DOI 10.1016/0021-8693(67)90043-9
Additional Information
- © Copyright 1976 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 219 (1976), 323-336
- MSC: Primary 16A18
- DOI: https://doi.org/10.1090/S0002-9947-1976-0419503-X
- MathSciNet review: 0419503