On the semiprimitivity of finitely generated algebras

Okniński, Jan

doi:10.1090/S0002-9939-2014-12187-3

On the semiprimitivity of finitely generated algebras
HTML articles powered by AMS MathViewer

by Jan Okniński PDF

Proc. Amer. Math. Soc. 142 (2014), 4095-4098 Request permission

Abstract:

Finitely generated associative algebras $A=K\langle a_{1},\ldots , a_{n}\rangle$ over a field $K$ defined by homogeneous relations are considered. If there exists an order on the associated free monoid $\mathrm {FM}_{n}$ of rank $n$ such that the set of normal forms of elements of $A$ is a regular language in $\mathrm {FM}_{n}$, then the algebra $A$ is semiprimitive provided that the associated monomial algebra is semiprime.

References

Jason P. Bell and Pinar Colak, Primitivity of finitely presented monomial algebras, J. Pure Appl. Algebra 213 (2009), no. 7, 1299–1305. MR 2497577, DOI 10.1016/j.jpaa.2008.11.039
Ferran Cedó and Jan Okniński, Gröbner bases for quadratic algebras of skew type, Proc. Edinb. Math. Soc. (2) 55 (2012), no. 2, 387–401. MR 2928501, DOI 10.1017/S0013091511000447
Mikhail V. Kochetov, V. T. Filippov, V. K. Kharchenko, and I. P. Shestakov (eds.), Dniester notebook: unsolved problems in the theory of rings and modules, Non-associative algebra and its applications, Lect. Notes Pure Appl. Math., vol. 246, Chapman & Hall/CRC, Boca Raton, FL, 2006, pp. 461–516. Translated from the 1993 Russian edition [MR1310114] by Murray R. Bremner and Mikhail V. Kochetov and edited by V. T. Filippov, V. K. Kharchenko and I. P. Shestakov. MR 2203726, DOI 10.1201/9781420003451.axb
Carl Faith, Algebra. II, Grundlehren der Mathematischen Wissenschaften, No. 191, Springer-Verlag, Berlin-New York, 1976. Ring theory. MR 0427349
G. Karpilovsky, The Jacobson radical of classical rings, Pitman Monographs and Surveys in Pure and Applied Mathematics, vol. 53, Longman Scientific & Technical, Harlow; copublished in the United States with John Wiley & Sons, Inc., New York, 1991. MR 1124405
Günter R. Krause and Thomas H. Lenagan, Growth of algebras and Gelfand-Kirillov dimension, Revised edition, Graduate Studies in Mathematics, vol. 22, American Mathematical Society, Providence, RI, 2000. MR 1721834, DOI 10.1090/gsm/022
Jan Krempa and Jan Okniński, Some examples of affine monoid algebras, Comm. Algebra 40 (2012), no. 1, 78–103. MR 2876291, DOI 10.1080/00927872.2010.524685
Gérard Lallement, Semigroups and combinatorial applications, Pure and Applied Mathematics, John Wiley & Sons, New York-Chichester-Brisbane, 1979. MR 530552
H. Li, Recognizing the semiprimitivity of N-graded algebras via Gröbner bases, preprint arXiv:1110.2248v1, to appear in Algebra Colloquium.
Jonas Månsson and Patrik Nordbeck, Regular Gröbner bases, J. Symbolic Comput. 33 (2002), no. 2, 163–181. MR 1879379, DOI 10.1006/jsco.2001.0500
Jonas Månsson and Patrik Nordbeck, A generalized Ufnarovski graph, Appl. Algebra Engrg. Comm. Comput. 16 (2005), no. 5, 293–306. MR 2233759, DOI 10.1007/s00200-005-0178-8
Jan Okniński, Semigroup algebras, Monographs and Textbooks in Pure and Applied Mathematics, vol. 138, Marcel Dekker, Inc., New York, 1991. MR 1083356
Jan Okniński, Structure of prime finitely presented monomial algebras, J. Algebra 320 (2008), no. 8, 3199–3205. MR 2450723, DOI 10.1016/j.jalgebra.2008.08.003
Donald S. Passman, The algebraic structure of group rings, Pure and Applied Mathematics, Wiley-Interscience [John Wiley & Sons], New York-London-Sydney, 1977. MR 0470211
Agata Smoktunowicz, Some results in noncommutative ring theory, International Congress of Mathematicians. Vol. II, Eur. Math. Soc., Zürich, 2006, pp. 259–269. MR 2275597
V. A. Ufnarovskij, Combinatorial and asymptotic methods in algebra [ MR1060321 (92h:16024)], Algebra, VI, Encyclopaedia Math. Sci., vol. 57, Springer, Berlin, 1995, pp. 1–196. MR 1360005, DOI 10.1007/978-3-662-06292-0_{1}
Efim Zelmanov, Some open problems in the theory of infinite dimensional algebras, J. Korean Math. Soc. 44 (2007), no. 5, 1185–1195. MR 2348741, DOI 10.4134/JKMS.2007.44.5.1185