×

zbMATH — the first resource for mathematics

Differentiably simple alternative algebras. (English. Russian original) Zbl 1283.17023
Algebra Logic 49, No. 5, 456-469 (2010); translation from Algebra Logika 49, No. 5, 670-689 (2010).
From the text: Differentiably simple algebras have been studied in several works (see, e.g., [E. C. Posner, Proc. Am. Math. Soc. 11, 337–343 (1960; Zbl 0103.26802)], [L. R. Harper jun., Trans. Am. Math. Soc. 100, 63–72 (1961; Zbl 0099.02203)]). Posner proved that every differentiably simple associative ring of characteristic 0 is prime, and it is simple if a minimal ideal is at hand. Differentiably simple associative commutative rings of characteristic \(p>0\) were described in [S. Yuan, Duke Math. J. 31, 623–630 (1964; Zbl 0145.27701)] under the assumption that the Jacobson radical is nilpotent. In [R. E. Block, Ann. Math. (2) 90, 433–459 (1969; Zbl 0216.07303)], differentiably simple rings in arbitrary characteristic with minimal ideals were classified in terms of simple rings.
In the present paper the author proves that a differentiably simple alternative nonassociative algebra \(A\) over a field of characteristic 0 is a prime algebra quadratic over its center \(Z(A)\) and is a finitely generated projective \(Z(A)\)-module.
Moreover, he shows that over a field of characteristic \(p > 0\), \(A\) appears as a tensor product of the center \(Z(A)\) and some Cayley-Dixon algebra over an extension of a base field. In this case, \(Z(A)\) is a differentiably simple (with respect to some family of derivations) associative commutative algebra.

MSC:
17D05 Alternative rings
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] E. C. Posner, ”Differentiably simple rings,” Proc. Am. Math. Soc., 11, No. 3, 337–343 (1960). · Zbl 0103.26802 · doi:10.1090/S0002-9939-1960-0113908-6
[2] L. R. Harper, Jun., ”On differentiably simple algebras,” Trans. Am. Math. Soc., 100, No. 1, 63–72 (1961). · Zbl 0099.02203 · doi:10.1090/S0002-9947-1961-0130250-3
[3] Shuen Yuan, ”Differentiably simple rings of prime characteristic,” Duke Math. J., 31, No. 4, 623–630 (1964). · Zbl 0145.27701 · doi:10.1215/S0012-7094-64-03161-8
[4] R. E. Block, ”Determination of the differentiably simple rings with a minimal ideal,” Ann. Math. (2), 90, No. 3, 433–459 (1969). · Zbl 0216.07303 · doi:10.2307/1970745
[5] K. A. Zhevlakov, A. M. Slin’ko, I. P. Shestakov, and A. I. Shirshov, Rings That Are Nearly Associative [in Russian], Nauka, Moscow (1978). · Zbl 0445.17001
[6] A. M. Slin’ko, ”Remark on radicals and differentiation of rings,” Sib. Math. Zh., 13, No. 6, 1395–1397 (1973).
[7] R. D. Schafer, An Introduction to Nonassociative Algebras, Academic Press, London (1966). · Zbl 0145.25601
[8] N. Jacobson, ”A Kronecker factorization theorem for Cayley algebras and the exceptional simple Jordan algebra,” Am. J. Math., 76, 447–452 (1954). · Zbl 0055.26502 · doi:10.2307/2372584
[9] G. Azumaya, ”On maximally central algebras,” Nagoya Math. J., 2, 119–150 (1951). · Zbl 0045.01103
[10] N. Bourbaki, Commutative Algebra, Hermann, Paris (1972).
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.