A note on the π-complemented algebras.
Abstract
π-complemented algebras are defined as those (not necessarily associative or unital) algebras such that each annihilator ideal is complemented by other annihilator ideal. Let BA denote the set of all idempotents of the extended centroid of a semiprime algebra A. We prove that if there is a maximal ideal P of BA such that PA ⊆ A then A and PA are two π-complemented algebras. As a consequence, we give a characterization of the π-complementation of the unitisation, and the multiplication ideal, of a semiprime algebra.
Keywords
Full Text:
PDFReferences
W. E. Baxter and W. S. Martindale III, Central closure of semiprime nonassociative rings. Comm. Algebra 7 (1979), 1103-1132.
K. I. Beidar, W. S. Martindale III, and A. V. Mikhalev, Rings with generalized identities. Marcel Dekker, New York, 1996.
M. Bresar, M.A. Chebotar, and W. S. Martindale III, Functional identities. Birkhauser Verlag, Basel-Boston-Berlin, 2007.
J. C. Cabello and M. Cabrera, Structure theory for multiplicatively semiprime algebras. J. Algebra 282 (2004), 386-421.
J. C. Cabello, M. Cabrera, and A. Fernández, π-Complemented algebras through pseudocomplemented lattices. Order 29 (2012), 463-
J. C. Cabello, M. Cabrera, and E. Nieto, Closed Prime ideals in Algebras with Semiprime Multiplication Algebras. Comm. Algebra 35 (2007), 4245- 4276.
J. C. Cabello, M. Cabrera, A. RodrÃguez, and R. Roura, A characterization of Ï€-complemented algebras. Comm. Algebra 41 (2013),
-3079.
J. C. Cabello, M. Cabrera, and R. Roura, π-complementation in the unitisation and multiplication algebras of a semiprime algebra.
Comm. Algebra 40(2012), 3507 - 3531.
J. C. Cabello, M. Cabrera, and R. Roura, Complementedly dense ideals. Decomposable algebras. J. Algebra Appl. Vol. 12, No. 7 (2013) 1350023 (27 pages). DOI: 10.1142/S0219498813500230
M. Cabrera and A. A. Mohammed, Extended centroid and central closure of multiplicatively semiprime algebras. Comm. Algebra 29 (2001), 1215-1233.
M. Ferrero, Closed and Prime Ideals in Free Centred Extensions. J. Algebra 148 (1992), 1-16.
M. Ferrero, Closed submodules of nomalizing bimodules over semiprime rings. Comm. Algebra 29 (2001), 1513-1550.
M. Ferrero and R. Wisbauer, Closure operations in module categories. Algebra Colloq. 3 (1996), 169-182.
Yu. P. Razmyslov, Identities of algebras and their representations. Translations of Math. Monographs, 138. AMS, 1994.
R. Wisbauer, Modules and algebras: bimodule structure and group actions on algebras. Pitman Monographs and Surveys in Pure and Applied Mathematics, 81. Longman, Harlow, 1996.
Refbacks
- There are currently no refbacks.
Copyright (c) 2016 Journal of Progressive Research in Mathematics
This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License.
Copyright © 2016 Journal of Progressive Research in Mathematics. All rights reserved.
ISSN: 2395-0218.
For any help/support contact us at editorial@scitecresearch.com, jprmeditor@scitecresearch.com.