Abstract
For an ideal I of a nonassociative algebra A, the π-closure of I is defined by \(\overline{I} = {\rm Ann}({\rm Ann} (I))\), where Ann(I) denotes the annihilator of I, i.e., the largest ideal J of A such that IJ = JI = 0. An algebra A is said to be π-complemented if for every π-closed ideal U of A there exists a π-closed ideal V of A such that A = U ⊕ V. For instance, the centrally closed semiprime ring, and the AW ∗-algebras (or more generally, boundedly centrally closed C ∗-algebras) are π-complemented algebras. In this paper we develop a structure theory for π-complemented algebras by using and revisiting some results of the structure theory for pseudocomplemented lattices.
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Ambrose, W.: Structure theorems for a special class of Banach algebras. Trans. Am. Math. Soc. 57, 364–386 (1945)
Ara, P., Mathieu, M.: Local multipliers of C ∗-algebras. In: Springer Monographs in Mathematics. Springer, London (2003)
Beidar, K.I., Martindale III, W.S., Mikhalev, A.V.: Rings with Generalized Identities. Marcel Dekker, New York (1996)
Cabello, J.C., Cabrera, M.: Structure theory for multiplicatively semiprime algebras. J. Algebra 282, 386–421 (2004)
Cabello, J.C., Cabrera, M., Nieto, E.: ε-Complemented algebras. Preprint, University of Granada (2008)
Cohn, D.L.: Measure Theory. Birkhäuser, Boston (1980)
Dales, H.G.: Banach algebras and automatic continuity. In: London Math. Soc. Monographs New Series, vol. 24. Clarendon Press, Oxford (2000)
Draper, C., Fernández López, A., García, E., Gómez Lozano, M.: The socle of a nondegenerate Lie algebra. J. Algebra 319, 2372–2394 (2008)
Essannouni, H., Kaidi, A.: Goldie’s theorems for alternative rings. Proc. Am. Math. Soc. 121, 39–45 (1994)
Fernández López, A., García, E., Gómez Lozano, M.: An Artinian theory for Lie algebras. J. Algebra 319, 938–951 (2008)
Fernández López, A., García Rus, E.: On the socle of a noncommutative Jordan algebra. Manuscr. Math. 56, 269–278 (1986)
Fernández López, A., García Rus, E.: Nondegenerate Jordan algebras satisfying local Goldie conditions. J. Algebra 182, 52–59 (1996)
Fernández López, A., García Rus, E.: Algebraic lattices and nonassociative structures. Proc. Am. Math. Soc. 126, 3211–3221 (1998)
Fernández López, A., García Rus, E., Montaner, F.: Goldie theory for Jordan algebras. J. Algebra 248, 397–471 (2002)
Fernández López, A., Rodríguez Palacios, A.: A Wedderburn theorem for non-associative complete normed algebras. J. Lond. Math. Soc. 33, 328–338 (1986)
Fernández López, A., Tocón, M.: Pseudocomplemented semilattice, Boolean algebras and compatible products. J. Algebra 242, 60–91 (2001)
Finston, D.R.: On multiplication algebras. Trans. Am. Math. Soc. 293, 807–818 (1986)
Fountain, J., Gould, V.: Orders on regular rings with minimal condition for principal right ideals. Commun. Algebra 19, 1501–1527 (1991)
Frink, O.: Pseudo-complements in semilattices. Duke Math. J. 29, 505–514 (1969)
Glivenko, V.: Sur quelques points de la logique de M. Brouwer. Bull. Acad. Sci. Belgique 15, 183–188 (1929)
Gómez Lozano, M., Siles Molina, M.: Left quotient rings of alternative rings. J. Algebra Appl. 6, 71–102 (2007)
Grätzer, G.: General Lattice Theory, 2nd edn. New appendices by the author with B.A. Davey, R. Freese, B. Ganter, M. Greferath, P. Jipsen, H.A. Priestley, H. Rose, E.T. Schmidt, S.E. Schmidt, F. Wehrung, and R. Wille. Birkhäuser, Basel (1998)
Jacobson, N.: A note on nonassociative algebras. Duke Math. J. 3, 544–548 (1937)
Jacobson, N.: Lie Algebras. Interscience, New York (1962)
Jacobson, N.: Structure and Representations of Jordan Algebras. Colloquium Publications, vol. 39. Amer. Math. Soc., Providence (1968)
Kaplansky, I.: Normed algebras. Duke Math. J. 16, 399–418 (1949)
Katriñák, T.: A new proof of the Glivenko–Frink theorem. Bull. Soc. R. Sci. Liège 50, 171 (1981)
Lam, T.Y.: Lectures on Modules and Rings. Springer, New York (1999)
Lambek, J.: Lectures on Rings and Modules. Blaisdell, Waltham (1966)
Laustsen, N.J., Loy, R.J., Read, Ch.J.: The lattice of closed ideals in the Banach algebra of operators on certain Banach spaces. J. Funct. Anal. 214, 106–121 (2004). Erratum 220, 240–241 (2005)
Loos, O.: On the socle of a Jordan pair. Collect. Math. 40, 109–125 (1989)
Loos, O., Neher, E.: Complementation of inner ideals in Jordan pairs. J. Algebra 166, 255–295 (1994)
Sari, B., Schlumprecht, Th., Tomczak-Jaegermann, N., Troitdky, V.G.: On norm closed ideals in L(ℓ p , ℓ q ). Stud. Math. 179, 239–262 (2007)
Schafer, R.D.: An Introduction to Nonassociative Algebras. Academic, New York (1966)
Schafer, R.D.: On structurable algebras. J. Algebra 92, 400–412 (1985)
Tomiuk, B.J.: Structure theory for complementd Banach algebras. Can. J. Math. 14, 651–659 (1962)
White, K.: Amenability and ideal structure of some Banach sequence algebras. J. Lond. Math. Soc. 68, 444–460 (2003)
Yood, B.: Closed prime ideals in topological rings. Proc. Lond. Math. Soc. 24, 307–323 (1972)
Zelmanov, E.I.: Goldie’s theorem for Jordan algebras. Sib. Math. J. 28, 44–52 (1987)
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J.C. Cabello and M. Cabrera were supported by the MICINN and Fondos FEDER, MTM2009-12067, and, in addition, by the Junta de Andalucía Grant FQM290. A. Fernández López was supported by the MEC and Fondos FEDER, MTM2007-61978 and MTM2010-19482.
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Cabello, J.C., Cabrera, M. & Fernández López, A. π-Complemented Algebras Through Pseudocomplemented Lattices. Order 29, 463–479 (2012). https://doi.org/10.1007/s11083-011-9214-4
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DOI: https://doi.org/10.1007/s11083-011-9214-4
Keywords
- Pseudocomplemented lattices
- Semiprime algebras
- Closure operations
- Complemented algebras
- Decomposable algebras