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The structure of alternative superbimodules. (English. Russian original) Zbl 0840.17030

Algebra Logic 33, No. 6, 386-397 (1994); translation from Algebra Logika 33, No. 6, 689-707 (1994).
The main purpose of the paper under review is to describe the alternative superbimodules of semisimple alternative superalgebras over an algebraically closed field of characteristic different from 2 and 3. The main results are: (i) If the superalgebra does not contain a supersubalgebra of type \((M_1 (F), M_1 (F))\) in its decomposition into prime ideals, then the superbimodules are merely alternative bimodules and their description can be obtained from R. D. Schafer [Trans. Am. Math. Soc. 72, 1-17 (1952; Zbl 0046.03503)]. (ii) For the superalgebra \((M_1 (F), M_1(F))\) the author constructs a family of superbimodules which gives a complete list of the irreducible superbimodules.
Reviewer: V.Drensky (Sofia)

MSC:

17D05 Alternative rings
17A70 Superalgebras

Citations:

Zbl 0046.03503
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References:

[1] R. D. Schafer, ?Representation of alternative algebras,?Trans. Am. Math. Soc.,72, No. 1, 1-17 (1952). · Zbl 0046.03503 · doi:10.1090/S0002-9947-1952-0045101-X
[2] N. Jacobson, ?Structure of alternative and Jordan bimodules,?Osaka J. Math.,6, No. 1, 1-71 (1954). · Zbl 0059.02902
[3] N. A. Pisarenko, ?The Wedderburn decomposition in finite-dimensional alternative superalgebras,?Algebra Logika,32, No. 4, 428-440 (1993). · Zbl 0840.17004
[4] R. S. Pierce,Associative Algebras, Springer, New York (1982). · Zbl 0497.16001
[5] E. I. Zelmanov and I. P. Shestakov, ?Prime alternative superalgebras and nilpotency of the radical of the free nilpotent algebra,?Izv. Akad. Nauk SSSR, Ser. Mat.,54, No. 4, 676-693 (1990). · Zbl 0713.17020
[6] A. R. Kemer, ?Identities of associative algebras are finitely based,?Algebra Logika,26, No. 5, 597-641 (1987).
[7] The Dnestrov Notebook, Unsolved Problems in the Theory of Rings and Modules, 4th edn., Novosibirsk (1993).
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