AI 中文总结
本文研究特征非2的域上Cayley-Dickson代数的零因子，明确双交替零因子的性质与显式形式，刻画各向异性范数情形下零因子图的结构，得到C等价准则及零化子维数的可能取值。
AI 中文摘要
研究任意域𝔽（特征char 𝔽≠2）上的Cayley-Dickson代数的零因子，证明其分量两两强交替且范数非零的零因子，在Cayley-Dickson代数的零因子图中构成六边形结构；建立至少一个分量范数非零的双交替零因子的性质，得到其零化子、正交化子与中心化子的显式形式；描述具有各向异性范数的Cayley-Dickson代数中零因子的性质，证明该情形下零因子图中的有向六边形可扩展为正交图中的无向双六边形；得到具有各向异性范数的Cayley-Dickson代数元素的C等价准则，同时考察Cayley-Dickson代数元素的零化子维数的可能取值。
英文摘要
Zero divisors of Cayley-Dickson algebras over an arbitrary field $\mathbb{F}$, $\mathrm{char} \, \mathbb{F} \neq 2$, are studied. It is shown that zero divisors, whose components alternate strongly pairwise and have nonzero norm, form hexagonal structures in the zero divisor graph of a Cayley-Dickson algebra. The properties of doubly alternative zero divisors, at least one of whose components has nonzero norm, are established, and an explicit form of their annihilators, orthogonalizers, and centralizers is obtained. The properties of zero divisors in Cayley-Dickson algebras with anisotropic norm are described, and it is shown that in this case directed hexagons in the zero divisor graph can be extended to undirected double hexagons in the orthogonality graph. A criterion of $C$-equivalence for elements of Cayley-Dickson algebras with anisotropic norm is obtained. Possible values of dimension for annihilators of elements of Cayley-Dickson algebras are considered.
Journal refS. Zhilina, On doubly alternative zero divisors in Cayley-Dickson algebras, J. Math. Sci. 272(4) (2023) 496-518
DOI：10.1007/s10958-023-06444-8
