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.

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.

Reviewer: Olaf Ninnemann (Berlin)

##### MSC:

17D05 | Alternative rings |

##### Keywords:

alternative algebra; Cayley-Dixon ring; Cayley-Dixon algebra; derivation; differentiably simple algebra
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\textit{A. A. Popov}, Algebra Logic 49, No. 5, 456--469 (2010; Zbl 1283.17023); translation from Algebra Logika 49, No. 5, 670--689 (2010)

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##### 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). |

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