Hostname: page-component-78c5997874-t5tsf Total loading time: 0 Render date: 2024-11-16T23:22:48.992Z Has data issue: false hasContentIssue false
  • English
  • Français

Quasi-Duality, Linear Compactness and Morita Duality for Power Series Rings

Published online by Cambridge University Press:  20 November 2018

Weimin Xue
Weimin Xue*
Affiliation:
Department of Mathematics, Fujian Normal University, Fuzhou, Fujian 350007, People's Republic of China
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.

AS a generalization of Morita duality, Kraemer introduced the notion of quasi-duality and showed that each left linearly compact ring has a quasi-duality. Let R be an associative ring with identity and R[[x]] the power series ring. We prove that (1) R[[x]] has a quasi-duality if and only if R has a quasi-duality; (2) R[[x]] is left linearly compact if and only if R is left linearly compact and left noetherian; and (3) R[[x]] has a Morita duality if and only if R is left noetherian and has a Morita duality induced by a bimodule RUS such that S is right noetherian.

Keywords

16D9016P40
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 39 , Issue 2 , 01 June 1996 , pp. 250 - 256
Copyright
Copyright © Canadian Mathematical Society 1996

References

1. Anderson, F. W. and Fuller, K. R., Rings and Categories of Modules, 2nd edition, Springer-Verlag, New York, 1992.Google Scholar
2. Anh, P. N., Morita duality for commutative rings, Comm. Algebra 18(1990), 1781–1788.Google Scholar
3. Azumaya, G., A duality theory for injective modules, Amer. J. Math. 81(1959), 249–278.Google Scholar
4. Dischinger, F. and Mûller, W., Left PF is not right PF, Comm. Algebra 14( 1986), 1223–1227.Google Scholar
5. Kraemer, J., Characterizations of the Existence of (Quasi-) Self duality for Complete Tensor Rings, Algebra Berichte 56, Verlag Reinhard Fischer, Munchen, 1987.Google Scholar
6. McKerrow, A. S., On the injective dimension of modules of power series, Quart. J. Math. Oxford 25(1974), 359–368.Google Scholar
7. Menini, C., Jacobson s conjecture, Morita duality and related questions, J. Algebra 103(1986), 634–655.Google Scholar
8. Morita, K., Duality for modules and its applications to the theory of rings with minimum condition, Tokyo Kyoiku Daigaku, Ser A6(1958), 83–142.Google Scholar
9. Muller, B. J., Linear compactness and Morita duality, J. Algebra 16(1970), 60–66.Google Scholar
10. Muller, B. J., Duality theory for linearly topologized modules, Math. Z. 119(1971), 63—74.Google Scholar
11. Vâmos, P., Rings with duality, Proc. London Math. Soc. 35(1977), 275–289.Google Scholar
12. Varadarajan, K., A generalization ofHilbert basis theorem, Comm. Algebra 10(1982), 2191–2204.Google Scholar
13. Xue, Weimin, Rings with Morita Duality, Lect. Notes Math. 1523, Springer-Verlag, Berlin, 1992.Google Scholar
14. Xue, Weimin, Morita duality and some kinds of ring extensions, Algebra Collq. 1(1994), 77–84.Google Scholar