Alternator and associator ideal algebras

Authors:
Irvin Roy Hentzel, Giulia Maria Piacentini Cattaneo and Denis Floyd

Journal:
Trans. Amer. Math. Soc. **229** (1977), 87-109

MSC:
Primary 17E05

DOI:
https://doi.org/10.1090/S0002-9947-1977-0447361-7

MathSciNet review:
0447361

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Abstract: If *I* is the ideal generated by all associators, , it is well known that in any nonassociative algebra . We examine nonassociative algebras where . Such algebras include algebras, Lie algebras, and, as we show, a large number of associator dependent algebras. An alternator is an associator of the type (*a, a, b*), (*a, b, a*), (*b, a, a*). We next study algebras where the additive span of all alternators is an ideal. These include all algebras where as well as alternative algebras. The last section deals with prime, right alternative, alternator ideal algebras satisfying an identity of the form for fixed . With two exceptions, if this algebra has an idempotent *e* such that , then the algebra is alternative. All our work deals with algebras with an identity element over a field of characteristic prime to 6. All our containment relations are given by identities.

**[1]**M. Burrow,*Representation theory of finite groups*, Academic Press, New York, 1965. MR**38**#250. MR**0231924 (38:250)****[2]**H. Çelik and D. Outcalt,*Power-associativity of antiflexible rings*(unpublished).**[3]**S. Getu and D. Rodabaugh,*Generalizing alternative rings*, Comm. Algebra**2**(1974), 35-81. MR**50**#4682. MR**0352195 (50:4682)****[4]**I. Hentzel,*The characterization of**rings*, J. Algebra**30**(1974), 236-258. MR**0417248 (54:5305)****[5]**I. R. Hentzel and G. M. Piacentini Cattaneo,*A note on generalizing alternative rings*, Proc. Amer. Math. Soc.**55**(1976), 6-8. MR**0393157 (52:13967)****[6]**M. Humm Kleinfeld,*On a class of right alternative rings without nilpotent ideals*, J. Algebra**5**(1967), 164-174. MR**35**#2939. MR**0212064 (35:2939)****[7]**E. Kleinfeld,*Right alternative rings*, Proc. Amer. Math. Soc.**4**(1953), 939-944. MR**15**, 595. MR**0059888 (15:595j)****[8]**E. Kleinfeld, F. Kosier, J. M. Osborn, and D. Rodabaugh,*The structure of associator dependent rings*, Trans. Amer. Math. Soc.**110**(1964), 473-483. MR**28**#1221. MR**0157993 (28:1221)****[9]**A. Thedy,*Right alternative rings*, J. Algebra**37**(1975), 1-43. MR**0384888 (52:5758)**

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1977-0447361-7

Keywords:
Associator ideal,
right alternative,
associator dependent

Article copyright:
© Copyright 1977
American Mathematical Society