Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS

   
Mobile Device Pairing
Green Open Access
Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

The Wedderburn principal theorem for a generalization of alternative algebras


Author: Harry F. Smith
Journal: Trans. Amer. Math. Soc. 198 (1974), 139-154
MSC: Primary 17A30
MathSciNet review: 0352187
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: A generalized alternative ring I is a nonassociative ring $ R$ in which the identities $ (wx,y,z) + (w,x,[y,z]) - w(x,y,z) - (w,y,z)x;([w,x],y,z) + (w,x,yz) - y(w,x,z) - (w,x,y)z$; and $ (x,x,x)$ are identically zero. It is here demonstrated that if $ A$ is a finite-dimensional algebra of this type over a field $ F$ of characteristic # 2, 3, then $ A$ a nilalgebra implies $ A$ is nilpotent.

A generalized alternative ring II is a nonassociative ring $ R$ in which the identities $ (wx,y,z) + (w,x,[y,z]) - w(x,y,z) - (w,y,z)x$ and $ (x,y,x)$ are identically zero. Let $ A$ be a finite-dimensional algebra of this type over a field $ F$ of characteristic # 2. Then it is here established that (1) $ A$ a nilalgebra implies $ A$ is nilpotent; (2) $ A$ simple with no nonzero idempotent other than 1 and $ F$ algebraically closed imply $ A$ itself is a field; and (3) the standard Wedderburn principal theorem is valid for $ A$.


References [Enhancements On Off] (What's this?)

  • [1] A. A. Albert, Structure of algebras, Amer. Math. Soc. Colloq. Publ., vol. 24, Amer. Math. Soc., Providence, R. I., 1939. MR 1, 99. MR 0123587 (23:A912)
  • [2] -, Power-associative rings, Trans. Amer. Math. Soc. 64 (1948), 552-593. MR 10, 349. MR 0027750 (10:349g)
  • [3] N. Jacobson, Structure and representations of Jordan algebras,Amer. Math. Soc. Colloq. Publ., vol. 39, Amer. Math. Soc., Providence, R. I., 1968. MR 40 #4330. MR 0251099 (40:4330)
  • [4] E. Kleinfeld, Generalization of alternative rings. I, J. Algebra 18 (1971), 304-325. MR 43 #308. MR 0274545 (43:308)
  • [5] -, Generalization of alternative rings. II, J. Algebra 18 (1971), 326-339. MR 43 #308. MR 0274545 (43:308)
  • [6] E. Kleinfeld and L. A. Kokoris, Flexible algebras of degree one, Proc. Amer. Math. Soc. 13 (1962), 891-893. MR 25 #5088. MR 0141691 (25:5088)
  • [7] K. McCrimmon, Structure and representations of noncommutative Jordan algebras, Trans. Amer. Math. Soc. 121 (1966), 187-199. MR 32 #5700. MR 0188261 (32:5700)
  • [8] D. J. Rodabaugh, On the Wedderburn principal theorem, Trans. Amer. Math. Soc. 138 (1969), 343-361. MR 0330240 (48:8578)
  • [9] R. D. Schafer, The Wedderburn principal theorem for alternative algebras, Bull. Amer. Math. Soc. 55 (1949), 604-614. MR 10, 676. MR 0029895 (10:676g)
  • [10] -, Noncommutative Jordan algebras of characteristic 0, Proc. Amer. Math. Soc. 6 (1955), 472-475. MR 17, 10. MR 0070627 (17:10c)
  • [11] -, On noncommutative Jordan algebras, Proc. Amer. Math. Soc. 9 (1958), 110-117. MR 21 #2677.
  • [12] -, Restricted noncommutative Jordan algebras of characteristic $ p$, Proc. Amer. Math. Soc. 9 (1958), 141-144. MR 21 #2678. MR 0103915 (21:2678)
  • [13] -, An introduction to non-associative algebras, Pure and Appl. Math., vol. 22, Academic Press, New York, 1966. MR 35 #1643.
  • [14] -, Standard algebras, Pacific J. Math. 29 (1969), 203-223. MR 39 #5647. MR 0244332 (39:5647)
  • [15] -, Generalized standard algebras, J. Algebra 12 (1969), 386-417. MR 44 #268. MR 0283035 (44:268)
  • [16] H. F. Smith, Ph.D. dissertation, University of Iowa, Iowa City, 1972.

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 17A30

Retrieve articles in all journals with MSC: 17A30


Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1974-0352187-6
PII: S 0002-9947(1974)0352187-6
Keywords: Power-associative, noncommutative Jordan, generalized alternative rings I and II, alternative ring, finite-dimensional, nil, solvable, nilpotent, simple, degree one, algebraically closed field, nodal algebra, Albert decomposition, Peirce decomposition, Penico solvable, Wedderburn principal theorem, nil radical, separable
Article copyright: © Copyright 1974 American Mathematical Society