Proper identities of finitely generated commutative alternative algebras

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• 作者：Sergey V. Pchelintsev ^{pchelinzev@mail.ru" class="auth_mail" title="E-mail the corresponding author}
• 关键词：17D05 ; 17D10 ; 17C05 ; 17A70
• 刊名：Journal of Algebra
• 出版年：2017
• 出版时间：15 January 2017
• 年：2017
• 卷：470
• 期：Complete
• 页码：425-440
• 全文大小：364 K

文摘

We prove that every proper polynomial of degree at least 869316301880&_mathId=si1.gif&_user=111111111&_pii=S0021869316301880&_rdoc=1&_issn=00218693&md5=eea8e0af3be6507c6a743dab1005d87e" title="Click to view the MathML source">2n−2$2n-2$ is an identity of commutative alternative algebra of rank 869316301880&_mathId=si2.gif&_user=111111111&_pii=S0021869316301880&_rdoc=1&_issn=00218693&md5=b89774e999cb86661ee795136b29fc35" title="Click to view the MathML source">n⩾3$n⩾3$. Using this we deduce that every commutative alternative algebra of rank n   with the identity 869316301880&_mathId=si3.gif&_user=111111111&_pii=S0021869316301880&_rdoc=1&_issn=00218693&md5=a98cd3589e1e3a26ce836c50ad409806" title="Click to view the MathML source">x³=0$x^3=0$ is nilpotent of index at most 869316301880&_mathId=si23.gif&_user=111111111&_pii=S0021869316301880&_rdoc=1&_issn=00218693&md5=adbcfde9d0baa892ebc335ec88f4c249" title="Click to view the MathML source">4n−2$4n-2$. We also prove that the index of nilpotency of the associator ideal in the free commutative alternative algebra of rank 869316301880&_mathId=si2.gif&_user=111111111&_pii=S0021869316301880&_rdoc=1&_issn=00218693&md5=b89774e999cb86661ee795136b29fc35" title="Click to view the MathML source">n⩾3$n⩾3$ is equal to 869316301880&_mathId=si312.gif&_user=111111111&_pii=S0021869316301880&_rdoc=1&_issn=00218693&md5=ec232d9fb24caacdc5d8f9e358544c8b">869316301880-si312.gif">$\left[\frac{2n}{3}\right]$.