Chains of Prime Ideals and Primitivity of \(\mathbb {Z}\) -Graded Algebras

详细信息    查看全文

• 作者：Be鈥檈ri Greenfeld ; Andr茅 Leroy ; Agata Smoktunowicz…
• 关键词：Graded algebras ; Primitive rings ; Semiprimitive rings ; Brown ; McCoy radical ; Chains of prime ideals ; GK dimension ; Growth of algebra ; 16W50 ; 16P90
• 刊名：Algebras and Representation Theory
• 出版年：2015
• 出版时间：June 2015
• 年：2015
• 卷：18
• 期：3
• 页码：777-800
• 全文大小：370 KB
• 参考文献：1.Amberg, B., Kazarin, L.: Nilpotent p-algebras and factorized p-groups, Groups St. Andrews (2005)
  2.Artin, M.: Some problems on three-dimensional graded domains, Representation theory and algebraic geometry. Lond. Math. Soc. Lect. Note Ser. 238, 1鈥?9 (1995). Cambridge Univ. PressMathSciNet
  3.Bartholdi, L.: Branch rings, thinned rings, tree enveloping rings. Israel J. Math. 154, 93鈥?39 (2006)MATH MathSciNet View Article
  4.Bartholdi, L.: Branch rings, thinned rings, tree enveloping rings - Erratum. Israel J. Math. 193, 507鈥?08 (2013)MATH MathSciNet View Article
  5.Beidar, K.I., Fong, Y., Puczy艂owski, E.R.: Polynomial rings over nil rings cannot be homomorphically mapped onto rings with nonzero idempotents. J. Algebra 238, 389鈥?99 (2001)MATH MathSciNet View Article
  6.Beidar, K.I., Martindale III, W.S., Mikhalev, A.V.: Rings with Generalized Identities. Marcel Dekker, New York (1996)MATH
  7.Bell, J.P., Smoktunowicz, A.: Extended centres of finitely generated prime algebras. J. Algebra 319, 414鈥?31 (2008)MATH MathSciNet View Article
  8.Bell, J.P., Smoktunowicz, A.: The prime spectrum of algebras of quadratic growth. Comm. Algebra 38, 332鈥?45 (2009)MathSciNet View Article
  9.Bell, J.P., Smoktunowicz, A.: Rings of differential operators on curves. Israel J. Math. 192, 297鈥?10 (2012)MATH MathSciNet View Article
  10.Jason, P., Bell, A.: dichotomy result for prime algebras of Gelfand-Kirillov dimension two. J. Algebra 324, 831鈥?40 (2010)MATH MathSciNet View Article
  11.Chebotar, M., Ke, W.F., Lee, P.H., Puczy艂owski, E.R.: A note on polynomial rings over nil rings, in: Modules and Comodules, in: Trends Math, pp 169鈥?72. Birkhuser Verlag, Basel (2008)
  12.Chin, W.: Declan Quinn, Rings graded by polycyclic-by-finite groups. Proc. Amer. Math. Soc. 102(2), 235鈥?41 (1988)MATH MathSciNet View Article
  13.Ferrero, M., Wisbauer, R.: Unitary strongly prime rings and radicals, J. Pure Appl. Algebra 181, 209鈥?26 (2003)MATH MathSciNet View Article
  14.Greenfeld, B., Rowen, L.H., Vishne, U.: Union of chains of primes. arXiv: http://鈥媋rxiv.鈥媜rg/鈥媋bs/鈥?408.鈥?892
  15.Hartmut, L.: Upper central chains in rings, to appear in Comm. Algebra
  16.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)
  17.Krempa, J.: Logical connections among some open problems concerning nil rings. Fund. Math. 76, 121鈥?30 (1972)MathSciNet
  18.Lanski, Ch., Resco, R., Small, L.: On the primitivity of prime rings. J. Algebra 59, 395鈥?98 (1979)MATH MathSciNet View Article
  19.Lam, T.Y.: A first course in noncommutative rings. Springer-Verlag, New York (1991)MATH View Article
  20.Lee, P.H., Puczy艂owski, E.R.: On prime ideals and radicals of polynomial rings and graded rings. J. Pure Appl. Algebra 218, 323鈥?32 (2014)MATH MathSciNet View Article
  21.N膬st膬sescu, C., Van Oystaeyen, F.: Dimensions of ring theory. Reidel, Dordrecht (1987)MATH View Article
  22.Passman, D.: Prime ideals in normalizing extensions. J. Algebra 73, 556鈥?72 (1981)MATH MathSciNet View Article
  23.Passman, D.: Infinite Crossed Products. Dover Publications Inc, Mineola (2013)
  24.Posner, E.C.: Prime rings satisfying a polynomial identity. Proc. Amer. Math. Soc. 11, 180鈥?83 (1960)MATH MathSciNet View Article
  25.Procesi, C.: Rings with polynomial identities. Marcel Dekker, Inc, New York (1973)MATH
  26.Puczy艂owski, E.R., Smoktunowicz, A.: On maximal ideals and the Brown-McCoy radical of polynomial rings. Comm. Algebra 26, 2473鈥?482 (1998)MATH MathSciNet View Article
  27.Rowen, L.H.: Some results on the center of a ring with polynomial identity. Bull. Amer. Math. Soc. 79, 219鈥?23 (1973)MATH MathSciNet View Article
  28.Rowen, L.H.: Polynomial identities in ring theory (1980)
  29.Small, L.W., Warfield Jr, R.B.: Prime affine algebras of Gelfand-Kirillov dimension one. J. Algebra 91, 384鈥?89 (1984)View Article
  30.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鈥?40. Kluwer Acad. Publ., Dordrecht (2003)
  31.Smoktunowicz, A., Young, A.A.: Jacobson radical algebras with quadratic growth. Glasg. Math. J. 55, 135鈥?47 (2013)MathSciNet View Article
  32.Smoktunowicz, A.: A note on nil and Jacobson radicals in graded rings. J. Algebra Appl. 13, 8 (2014)MathSciNet View Article
  33.Smoktunowicz, A.: A simple nil ring exists. Comm. Algebra 30, 27鈥?9 (2002)MATH MathSciNet View Article
  
• 作者单位：Be鈥檈ri Greenfeld (1)
  Andr茅 Leroy (2)
  Agata Smoktunowicz (3)
  Micha艂 Ziembowski (4)
  
  1. Department of Mathematics, Bar Ilan University, Ramat Gan, 5290002, Israel
  2. Facult茅 Jean Perrin, Rue J. Souvraz, Universit茅 d鈥橝rtois, 62300, Lens, France
  3. Maxwell Institute for Mathematical Sciences, School of Mathematics, University of Edinburgh, JCM Building, Kings Buildings, Mayfield Road, Edinburgh, EH9 3JZ, Scotland, UK
  4. Faculty of Mathematics and Information Science, Warsaw University of Technology, 00-662, Warsaw, Poland
  
• 刊物主题：Commutative Rings and Algebras; Associative Rings and Algebras; Non-associative Rings and Algebras;
• 出版者：Springer Netherlands
• ISSN：1572-9079

文摘

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,鈭? 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 鈮? 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.