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ISSN 1088-6850(e) ISSN 0002-9947(p)

Polynomial identities in nil-algebras

Author(s): Elena V. Aladova; Alexei N. Krasilnikov
Journal: Trans. Amer. Math. Soc. 361 (2009), 5629-5646.
MSC (2000): Primary 16R10
Posted: June 23, 2009
MathSciNet review: 2529907
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Abstract: We prove that in associative algebras over a field of characteristic $p \ge 3$ the polynomial identity $x^{2p}=0$ is not Specht. To prove this we construct a non-finitely based system of polynomial identities which contains the identity $x^{2p}=0$ . We also give an example of a non-Specht polynomial identity of degree in unital associative -algebras.

References:

1.: E. V. Aladova, A non-finitely based variety of nil-algebras over a field of characteristic . (Russian) Algebra and linear optimization (Ekaterinburg, 2002), Ross. Akad. Nauk Ural. Otdel., Inst. Mat. Mekh., Ekaterinburg, 2002, pp. 5-11. MR 1847556 (2003e:16026)
2.: Yu. A. Bahturin, Identical relations in Lie algebras, VNU Science Press, Utrecht, 1987. MR 886063 (88f:17032)
3.: Yu. A. Bakhturin, A. Yu. Ol'shanskii, Identities, Algebra, II. Encyclopaedia of Mathematical Sciences, Current Problems in Mathematics, Fundamental Directions, vol. 18, Springer-Verlag, Berlin, 1991, pp. 117-240. MR 1121267 (92b:00057)
4.: A. Ya. Belov, On non-Specht varieties (Russian), Fundam. Prikl. Mat. 5 (1999), 47-66. MR 1799544 (2001k:16040)
5.: A. Ya. Belov, Counterexamples to the Specht problem (Russian). Mat. Sb. 191 (2000), 13-24. English translation in Sb. Math. 191 (2000), 329-340. MR 1773251 (2001g:16043)
6.: L. A. Bokyt', I. V. L'vov, V. K. Harchenko, Noncommutative rings, Algebra, II. Encyclopaedia of Mathematical Sciences, Current Problems in Mathematics, Fundamental Directions, vol. 18, Springer-Verlag, Berlin, 1991, pp. 5-116. MR 1121267 (92b:00057)
7.: V. Drensky, Free algebras and PI-algebras. Springer-Verlag Singapore, Singapore, 2000. MR 1712064 (2000j:16002)
8.: E. Formanek, The Nagata-Higman theorem, Acta Appl. Math. 21 (1990), 185-192. MR 1085778 (92d:15023)
9.: A. V. Grishin, Examples of -spaces and -ideals of characteristic without the finite basis property (Russian). Fundam. Prikl. Mat. 5 (1999), 101-118. MR 1799541 (2002a:16028)
10.: A. V. Grishin, On non-Spechtianness of the variety of associative rings that satisfy the identity $x^{32}=0$ , Electron. Res. Announc. Amer. Math. Soc. 6 (2000), 50-51 (electronic). MR 1777855
11.: C. K. Gupta, A. N. Krasilnikov, A non-finitely based system of polynomial identities which contains the identity , Quart. J. Math. 53 (2002), 173-183. MR 1909509 (2003c:16031)
12.: G. Higman, On a conjecture of Nagata, Proc. Cambridge Phil. Soc. 52 (1956), 1-4. MR 0073581 (17:453e)
13.: A. Kanel-Belov, L. H. Rowen, Computational aspects of polynomial identities, A K Peters, Ltd., Wellesley, MA, 2005. MR 2124127 (2006b:16001)
14.: A. R. Kemer, Finite basability of identities of associative algebras (Russian), Algebra i Logika 26 (1987), 597-641. MR 985840 (90b:08008)
15.: M. Nagata, On the nilpotency of nil algebras. J. Math. Soc. Japan 4 (1953), 296-301. MR 0053088 (14:719g)
16.: L. H. Rowen, Ring theory, Academic Press, Inc., Boston, MA, 1991. MR 1095047 (94e:16001)
17.: V. V. Shchigolev, Examples of infinitely based -ideals (Russian), Fundam. Prikl. Mat. 5 (1999), 307-312. MR 1799533 (2001k:16044)
18.: V. V. Shchigolev, Examples of infinitely basable -spaces (Russian), Mat. Sb. 191 (2000), 143-160. English translation in Sb. Math. 191 (2000), 459-476. MR 1773258 (2001f:16044)
19.: V. V. Shchigolev, Infinitely based -spaces and -ideals, Ph.D. thesis, Moscow State University, 2002.

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