Abstract
In this paper we provide some results regarding affine, prime, \(\mathbb {Z}\)-graded algebras \(R=\bigoplus _{i\in \mathbb {Z}}R_{i}\) generated by elements with degrees 1,−1 and 0, with R 0 finite-dimensional. The results are as follows. These algebras have a classical Krull dimension when they have quadratic growth. If R k ≠0 for almost all k then R is semiprimitive. If in addition R has GK dimension less than 3 then R is either primitive or PI. The tensor product of an arbitrary Brown-McCoy radical algebra of Gelfand Kirillov dimension less than three and any other algebra is Brown-McCoy radical.
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Amberg, B., Kazarin, L.: Nilpotent p-algebras and factorized p-groups, Groups St. Andrews (2005)
Artin, M.: Some problems on three-dimensional graded domains, Representation theory and algebraic geometry. Lond. Math. Soc. Lect. Note Ser. 238, 1–19 (1995). Cambridge Univ. Press
Bartholdi, L.: Branch rings, thinned rings, tree enveloping rings. Israel J. Math. 154, 93–139 (2006)
Bartholdi, L.: Branch rings, thinned rings, tree enveloping rings - Erratum. Israel J. Math. 193, 507–508 (2013)
Beidar, K.I., Fong, Y., Puczyłowski, E.R.: Polynomial rings over nil rings cannot be homomorphically mapped onto rings with nonzero idempotents. J. Algebra 238, 389–399 (2001)
Beidar, K.I., Martindale III, W.S., Mikhalev, A.V.: Rings with Generalized Identities. Marcel Dekker, New York (1996)
Bell, J.P., Smoktunowicz, A.: Extended centres of finitely generated prime algebras. J. Algebra 319, 414–431 (2008)
Bell, J.P., Smoktunowicz, A.: The prime spectrum of algebras of quadratic growth. Comm. Algebra 38, 332–345 (2009)
Bell, J.P., Smoktunowicz, A.: Rings of differential operators on curves. Israel J. Math. 192, 297–310 (2012)
Jason, P., Bell, A.: dichotomy result for prime algebras of Gelfand-Kirillov dimension two. J. Algebra 324, 831–840 (2010)
Chebotar, M., Ke, W.F., Lee, P.H., Puczyłowski, E.R.: A note on polynomial rings over nil rings, in: Modules and Comodules, in: Trends Math, pp 169–172. Birkhuser Verlag, Basel (2008)
Chin, W.: Declan Quinn, Rings graded by polycyclic-by-finite groups. Proc. Amer. Math. Soc. 102(2), 235–241 (1988)
Ferrero, M., Wisbauer, R.: Unitary strongly prime rings and radicals, J. Pure Appl. Algebra 181, 209–226 (2003)
Greenfeld, B., Rowen, L.H., Vishne, U.: Union of chains of primes. arXiv: http://arxiv.org/abs/1408.0892
Hartmut, L.: Upper central chains in rings, to appear in Comm. Algebra
Krause, G.R., Lenagan, T.H.: Growth of algebras and GelfandKirillov dimension, revised edition, Grad. Stud. Math. vol. 22, Amer. Math. Soc., Providence, RI (2000)
Krempa, J.: Logical connections among some open problems concerning nil rings. Fund. Math. 76, 121–130 (1972)
Lanski, Ch., Resco, R., Small, L.: On the primitivity of prime rings. J. Algebra 59, 395–398 (1979)
Lam, T.Y.: A first course in noncommutative rings. Springer-Verlag, New York (1991)
Lee, P.H., Puczyłowski, E.R.: On prime ideals and radicals of polynomial rings and graded rings. J. Pure Appl. Algebra 218, 323–332 (2014)
Năstăsescu, C., Van Oystaeyen, F.: Dimensions of ring theory. Reidel, Dordrecht (1987)
Passman, D.: Prime ideals in normalizing extensions. J. Algebra 73, 556–572 (1981)
Passman, D.: Infinite Crossed Products. Dover Publications Inc, Mineola (2013)
Posner, E.C.: Prime rings satisfying a polynomial identity. Proc. Amer. Math. Soc. 11, 180–183 (1960)
Procesi, C.: Rings with polynomial identities. Marcel Dekker, Inc, New York (1973)
Puczyłowski, E.R., Smoktunowicz, A.: On maximal ideals and the Brown-McCoy radical of polynomial rings. Comm. Algebra 26, 2473–2482 (1998)
Rowen, L.H.: Some results on the center of a ring with polynomial identity. Bull. Amer. Math. Soc. 79, 219–223 (1973)
Rowen, L.H.: Polynomial identities in ring theory (1980)
Small, L.W., Warfield Jr, R.B.: Prime affine algebras of Gelfand-Kirillov dimension one. J. Algebra 91, 384–389 (1984)
Smoktunowicz, A.: R[x,y] is Brown-McCoy radical if R[x] is Jacobson radical, Proceedings of the Third International Algebra Conference (Tainan, 2002), pp. 235–240. Kluwer Acad. Publ., Dordrecht (2003)
Smoktunowicz, A., Young, A.A.: Jacobson radical algebras with quadratic growth. Glasg. Math. J. 55, 135–147 (2013)
Smoktunowicz, A.: A note on nil and Jacobson radicals in graded rings. J. Algebra Appl. 13, 8 (2014)
Smoktunowicz, A.: A simple nil ring exists. Comm. Algebra 30, 27–59 (2002)
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Presented by Paul Smith.
A. Smoktunowicz was supported by ERC Advanced grant Coimbra 320974 and M. Ziembowski was supported by the Polish National Science Centre grant UMO-2013/09/D/ST1/03669.
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Greenfeld, B., Leroy, A., Smoktunowicz, A. et al. Chains of Prime Ideals and Primitivity of \(\mathbb {Z}\)-Graded Algebras. Algebr Represent Theor 18, 777–800 (2015). https://doi.org/10.1007/s10468-015-9516-0
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DOI: https://doi.org/10.1007/s10468-015-9516-0
Keywords
- Graded algebras
- Primitive rings
- Semiprimitive rings
- Brown-McCoy radical
- Chains of prime ideals
- GK dimension
- Growth of algebra