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Projective Modules Over Central Separable Algebras

Published online by Cambridge University Press:  20 November 2018

F. R. DeMeyer
F. R. DeMeyer*
Affiliation:
Purdue University, Lafayette, Indiana
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In (2), M. Auslander and O. Goldman laid the foundations for the study of central separable algebras. For unexplained terminology and notation, see (2). Here we are interested in projective modules and the ideal structure of a central separable algebra A over some special commutative rings K. When K is a field, one consequence of Wedderburn's Theorem is that there is a unique (up to isomorphism) irreducible A-module. We show here that if K is a commutative ring with a finite number of maximal ideals (semi-local) and with no idempotents other than 0 and 1 or if K is the ring of polynomials in one variable over a perfect field, then there is a unique (up to isomorphism) indecomposable finitely generated projective A-module. An example in (3) shows that this result fails if one only assumes that K is a principal ideal domain.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 21 , 1969 , pp. 39 - 43
Copyright
Copyright © Canadian Mathematical Society 1969

References

1. Auslander, M. and Goldman, O., Maximal orders, Trans. Amer. Math. Soc. 97 (1960), 1–24.Google Scholar
2. Auslander, M. and Goldman, O., The Brauer group of a commutative ring, Trans. Amer. Math. Soc. 97 (1960), 367–409.Google Scholar
3. Bass, H., K-theory and stable algebra, Inst. Hautes Etudes Sci. Publ. Math. No. 22 (1964), 5–60.Google Scholar
4. Bourbaki, N., Modules et anneaux semi-simples (Hermann, Paris, 1958).Google Scholar
5. Chase, S., Harrison, D. K., and Rosenberg, A., Galois theory and Galois cohomology of commutative rings, Mem. Amer. Math. Soc. No. 52 (1965), II+79 pp.Google Scholar
6. Childs, L. N. and DeMeyer, F. R., On automorphisms of separable algebras, Pacific J. Math. 23 (1967), 25–34.Google Scholar
7. Curtis, C. W. and Reiner, I., Representation theory of finite groups and associative algebras (Interscience, New York, 1962).Google Scholar
8. DeMeyer, F. R., The Brauer group of some separably closed rings, Osaka Math. J. 3 (1966), 201–204.Google Scholar
9. Deuring, M., Zur Théorie der Idealklassen in algebraischen Funktionkörpern, Math. Ann. 106 (1932), 103–106.Google Scholar
10. Janusz, G. J., Separable algebras over commutative rings, Trans. Amer. Math. Soc. 122 (1966), 461–479.Google Scholar
11. Kanzaki, T., On commutor rings and Galois theory of separable algebras, Osaka Math. J. 1 (1964), 103–115.Google Scholar
12. Rosenberg, A. and Zelinsky, D., Automorphisms of separable algebras, Pacific J. Math. 11 (1957), 1109–1118.Google Scholar