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Primitive superalgebras with superinvolution. (English) Zbl 0915.16036

The author studies primitive associative superalgebras and develops their structure theory in the spirit of classical structure theory of (ordinary) associative algebras. In particular, he proves a superanalogue of the density theorem, gives a description of the artinian simple superrings as algebras of endomorphisms of a finite dimensional superspace over a division superalgebra and determines the isomorphisms between two artinian simple superrings. He also presents a complete list of central division superalgebras over a field in terms of the (ordinary) central division algebras over the same field. Then the author describes the primitive superrings with a minimal right superideal. An important application is for the case when the primitive superring has a superinvolution which is crucial for the study of finite dimensional central simple Jordan algebras. Another application is the description of simple superrings with superinvolution.
Reviewer: V.Drensky (Sofia)

MSC:

16W55 “Super” (or “skew”) structure
16D60 Simple and semisimple modules, primitive rings and ideals in associative algebras
16P20 Artinian rings and modules (associative rings and algebras)
16K20 Finite-dimensional division rings
16S50 Endomorphism rings; matrix rings
16W10 Rings with involution; Lie, Jordan and other nonassociative structures
17A70 Superalgebras
17C70 Super structures
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References:

[1] Albert, A.A., Structure of algebras, Amer. math. soc. colloq. publ., 24, (1939), Amer. Math. Soc Providence
[2] Coghlan, F.; Hoffman, P., Division graded algebras in the brauer – wall group, Canad. math. bull., 39, 21-24, (1996) · Zbl 0922.16009
[3] Cuenca Mira, J.A.; Martin, A.Garcia; Gonzalez, C.Martin, Prime associative superalgebras with non-zero socle, Algebras groups and geom., 11, 359-369, (1994) · Zbl 0828.17005
[4] Gomez-Ambrosi, C., On the simplicity of Hermitian superalgebras, Nova J. algebra geom., 3, 193-198, (1995) · Zbl 0873.17004
[5] Herstein, I.N., Rings with involution, Chicago lectures in math., (1976), Univ. of Chicago Press Chicago · Zbl 0343.16011
[6] Jacobson, N., Structure of rings, Amer. math. soc. colloq. publ., 37, (1956), Amer. Math. Soc Providence
[7] Kac, G., Classification of simpleZ, Comm. algebra, 5, 1375-1400, (1977) · Zbl 0367.17007
[8] Kac, G., Lie superalgebras, Adv. math., 26, 8-96, (1977) · Zbl 0366.17012
[9] M. L. Racine, E. I. Zelmanov, Simple Jordan superalgebras with semisimple even part
[10] Rowen, L.H., Ring theory, (1988), Academic Press San Diego/London · Zbl 0651.16002
[11] Wall, C.T.C., Graded Brauer groups, J. reine angew. math., 213, 187-199, (1963) · Zbl 0125.01904
[12] C. Draper, A. Elduque, Division superalgebras, in, Proceedings of the Malaga Conference · Zbl 0977.16022
[13] M. L. Racine, Associative superalgebras with superinvolution, in, Proceedings of the Malaga Conference · Zbl 0977.16021
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