Hostname: page-component-586b7cd67f-t7czq Total loading time: 0 Render date: 2024-11-28T08:28:28.448Z Has data issue: false hasContentIssue false
  • English
  • Français

Inertial subalgebras of algebras possessing finite automorphism groups

Part of: Local rings and semilocal rings

Published online by Cambridge University Press:  09 April 2009

Nicholas S. Ford
Nicholas S. Ford
Affiliation:
The Fayette Campus The Pennsylvania State UniversityUniontown, Pennsylvania 15401, USA
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Let R be a commutative ring with identity, and let A be a finitely generated R-algebra with Jacobson radical N and center C. An R-inertial subalgebra of A is a R-separable subalgebra B with the property that B+N=A. Suppose A is separable over C and possesses a finite group G of R-automorphisms whose restriction to C is faithful with fixed ring R. If R is an inertial subalgebra of C, necessary and sufficient conditions for the existence of an R-inertial subalgebra of A are found when the order of G is a unit in R. Under these conditions, an R-inertial subalgebra B of A is characterized as being the fixed subring of a group of R-automorphisms of A. Moreover, A ⋍ B ⊗R C. Analogous results are obtained when C has an R-inertial subalgebra S ⊃ R.

MSC classification

Secondary: 13H99: None of the above, but in this section
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 28 , Issue 3 , November 1979 , pp. 335 - 345
Copyright
Copyright © Australian Mathematical Society 1979

References

Azumaya, G. (1951), ‘On maximally central algebras’, Nagoya Math. J. 2, 119–150.Google Scholar
Bourbaki, N. (1962), ‘Algebre commutative’, Actualities Sci. Ind. No. 1290 (Hermann, Paris).Google Scholar
Childs, L. N. (1972), ‘On normal Azumaya algebras and the Teichmüller cocycle map’, J. Algebra 23, 1–17.Google Scholar
Childs, L. N. and DeMeyer, F. R. (1967), ‘On automorphisms of separable algebras’, Pacific J. Math. 23, 25–34.Google Scholar
DeMeyer, F. R. (1965), ‘Some notes on the general Galois theory of rings’, Osaka J. Math. 2, 117–127.Google Scholar
DeMeyer, F. R. and Ingraham, E. C. (1971), Separable algebras over a commutative ring (Springer-Verlag, New York).CrossRefGoogle Scholar
Dickson, L. (1960), Algebras and their arithmetic (Dover Publications, New York).Google Scholar
Eilenberg, S. and MacLane, S. (1948), ‘Cohomology and Galois theory, I. Normality of algebras and Teichmüller cocycle’, Trans. Am. Math. Soc. 64, 1–20.Google Scholar
Endo, S. and Watanabe, Y. (1967), ‘On separable algebras over a commutative ring’, Osaka J. Math. 4, 233–242.Google Scholar
Ford, N. S. (1976), ‘Inertial subalgebras of central separable algebras’, Proc. Amer. Math. Soc. 60, 39–44.Google Scholar
Ingraham, E. C. (1966), ‘Inertial subalgebras of algebras over commutative rings’, Trans. Amer. Math. Soc. 124, 77–93.Google Scholar
Malcev, A. (1942), ‘On the representation of an algebra as the direct sum of the radical and a semi-simple algebra’, C.R. URSS 36.Google Scholar
Pareigis, B. (1964), ‘Über normale, zentrale, separable Algebren und Amitsur Kohomologie’, Math. Ann. 154, 330–340.Google Scholar