Geodesic
Nearly finite-dimensional Jordan algebras
Algebra i logika, Tome 57 (2018) no. 5, pp. 522-546.

Voir la notice de l'article provenant de la source Math-Net.Ru

Nearly finite-dimensional Jordan algebras are examined. Analogs of known results are considered. Namely, it is proved that such algebras are prime and nondegenerate. It is shown that the property of being nearly finite-dimensional is preserved in passing from an alternative algebra to an adjoint Jordan algebra. A similar result is established for associative nearly finite-dimensional algebras with involution. It is stated that a nearly finitedimensional Jordan PI-algebra with unity either is a finite module over a nearly finite-dimensional center or is a central order in an algebra of a nondegenerate symmetric bilinear form. Also the following result holds: if a locally nilpotent ideal has finite codimension in a Jordan algebra with the ascending chain condition on ideals, then that algebra is finite-dimensional. In addition, E. Formanek's result in [Commun. Algebra, 1, No. 1 (1974), 79–86], which says that associative prime PI-rings with unity are embedded in a free module of finite rank over its center, is generalized to Albert rings.
Keywords: nearly finite-dimensional Jordan algebra, associative nearly finite-dimensional algebra with involution, nearly finite-dimensional Jordan PIalgebra with unity, Albert ring. 
@article{AL_2018_57_5_a1,
     author = {V. N. Zhelyabin and A. S. Panasenko},
     title = {Nearly finite-dimensional {Jordan} algebras},
     journal = {Algebra i logika},
     pages = {522--546},
     publisher = {mathdoc},
     volume = {57},
     number = {5},
     year = {2018},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/AL_2018_57_5_a1/}
}
TY  - JOUR
AU  - V. N. Zhelyabin
AU  - A. S. Panasenko
TI  - Nearly finite-dimensional Jordan algebras
JO  - Algebra i logika
PY  - 2018
SP  - 522
EP  - 546
VL  - 57
IS  - 5
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/AL_2018_57_5_a1/
LA  - ru
ID  - AL_2018_57_5_a1
ER  - 
%0 Journal Article
%A V. N. Zhelyabin
%A A. S. Panasenko
%T Nearly finite-dimensional Jordan algebras
%J Algebra i logika
%D 2018
%P 522-546
%V 57
%N 5
%I mathdoc
%U http://geodesic.mathdoc.fr/item/AL_2018_57_5_a1/
%G ru
%F AL_2018_57_5_a1
V. N. Zhelyabin; A. S. Panasenko. Nearly finite-dimensional Jordan algebras. Algebra i logika, Tome 57 (2018) no. 5, pp. 522-546. http://geodesic.mathdoc.fr/item/AL_2018_57_5_a1/

[1] J. Bell, J. Farina, C. Pendergrass-Rice, “Stably just infinite rings”, J. Algebra, 319:6 (2008), 2533–2544 | DOI | MR | Zbl

[2] D. S. Passman, W. V. Temple, “Representations of the Gupta-Sidki group”, Proc. Am. Math. Sos., 124:5 (1996), 1403–1410 | DOI | MR | Zbl

[3] D. R. Farkas, L. W. Small, “Algebras which are nearly finite dimensional and their identities”, Isr. J. Math., 127:1 (2002), 245–251 | DOI | MR | Zbl

[4] Z. Reichstein, D. Rogalski, J. J. Zhang, “Projectively simple rings”, Adv. Math., 203:2 (2006), 365–407 | DOI | MR | Zbl

[5] L. Bartholdi, “Branch rings, thinned rings, tree enveloping rings”, Isr. J. Math., 154:1 (2006), 93–139 ; “Erratum”, Isr. J. Math., 193:1 (2013), 507–508 | DOI | MR | Zbl | DOI | MR | Zbl

[6] J. Farina, C. Pendergrass-Rice, “A few properties of just infinite algebras”, Commun. Algebra, 35:5 (2007), 1703–1707 | DOI | MR | Zbl

[7] C. Pendergrass-Rice, Extending a theorem of Herstein, arXiv: 0710.5545v1[math.RA]

[8] I. N. Herstein, “On the Lie and Jordan rings of a simple associative ring”, Am. J. Math., 77:2 (1955), 279–285 | DOI | MR | Zbl

[9] A. Shalev, E. I. Zelmanov, “Narrow Lie algebras: A coclass theory and a characterization of the Witt algebra”, J. Algebra, 189:2 (1997), 294–331 | DOI | MR | Zbl

[10] N. Gavioli, V. Monti, C. M. Scoppola, “Just infinite periodic Lie algebras”, Finite groups 2003, Proc. Gainesville conf. on finite groups (Gainesville, FL, USA, March 6–12, 2003), In honour of J. Thompson to his 70th birthday, eds. Chat Yin Ho et al., Walter de Gruyter, Berlin, 2004, 73–85 | MR | Zbl

[11] J. S. Wilson, “Groups with every proper quotient finite”, Proc. Camb. Philos. Soc., 69:3 (1971), 373–391 | DOI | MR | Zbl

[12] R. Grigorchuk, P. Shumyatsky, “On just-infinite periodic locally soluble groups”, Arch. Math., 109:1 (2017), 19–27 | DOI | MR | Zbl

[13] A. S. Panasenko, “Pochti konechnomernye alternativnye algebry”, Matem. zametki, 98:5 (2015), 747–755 | DOI | MR | Zbl

[14] V. N. Zhelyabin, A. S. Panasenko, “Nil-idealy konechnoi korazmernosti v alternativnykh neterovykh algebrakh”, Matem. zametki, 101:3 (2017), 395–402 | DOI | MR | Zbl

[15] E. Formanek, “Noetherian PI-rings”, Commun. Algebra, 1:1 (1974), 79–86 | DOI | MR | Zbl

[16] Unsolved problems in group theory, The Kourovka notebook, No. 19, Sobolev Institute of Mathematics, Novosibirsk, 2018 http://www.math.nsc.ru/~alglog/19tkt.pdf

[17] K. A. Zhevlakov, A. M. Slinko, I. P. Shestakov, A. I. Shirshov, Koltsa, blizkie k assotsiativnym, Nauka, M., 1978 | MR

[18] N. Jacobson, Structure and representations of Jordan algebras, Colloq. Publ., 39, Am. Math. Soc. 1968, Providence, RI | MR | Zbl

[19] V. G. Skosyrskii, “O radikalakh iordanovykh algebr”, Sib. matem. zh., 29:2 (1988), 154–166 | MR | Zbl

[20] E. I. Zelmanov, “Absolyutnye deliteli nulya i algebraicheskie iordanovy algebry”, Sib. matem. zh., 23:6 (1982), 100–116 | MR | Zbl

[21] E. I. Zelmanov, “O pervichnykh iordanovykh algebrakh. II”, Sib. matem. zh., 24:1 (1983), 89–104 | MR | Zbl

[22] I. Kherstein, Nekommutativnye koltsa, Mir, M., 1972

[23] S. V. Pchelintsev, “Pervichnye alternativnye algebry, blizkie k kommutativnym”, Izv. RAN. Ser. matem., 68:1 (2004), 183–206 | DOI | MR | Zbl

[24] S. V. Pchelintsev, “Isklyuchitelnye pervichnye alternativnye algebry”, Sib. matem. zh., 48:6 (2007), 1322–1337 | MR | Zbl

[25] A. M. Slinko, “Radikaly iordanovykh kolets, svyazannykh s alternativnymi”, Matem. zametki, 16:1 (1974), 135–140 | MR | Zbl

[26] K. McCrimmon, “On Herstein's theorems relating Jordan and associative algebras”, J. Algebra, 13 (1969), 382–392 | DOI | MR | Zbl

[27] S. A. Amitsur, “Rings with involution”, Isr. J. Math., 6 (1968), 99–106 | DOI | MR | Zbl