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Trivial nuclear ideals of a free alternative algebra. (English. Russian original) Zbl 0603.17015
Algebra Logic 24, 455-471 (1985); translation from Algebra Logika 24, No. 6, 696-717 (1985).
Let \(\Phi\) be a commutative associative ring with 1 and containing 1/6, and let \(A_ k\) be the free alternative \(\Phi\)-algebra on k generators. It is well known that \(A_ 1\) and \(A_ 2\) do not have trivial ideals, and A. V. Il’tyakov [Algebra Logika 23, No.2, 136-158 (1984; Zbl 0588.17020)] has recently proved that this is also true for \(A_ 3\). M. Humm and E. Kleinfeld [J. Comb. Theory 2, 140-144 (1967; Zbl 0153.060)] proved that there do exist trivial ideals in \(A_ k\) for \(k\geq 4\). In [Algebra Logika 21, No.1, 84-107 (1982; Zbl 0506.17009)] the author actually constructed such ideals in \(A_ k\) for \(k\geq 5\), and in [Algebra Logika 22, No.2, 182-197 (1983; Zbl 0542.17010)] he constructed trivial nuclear ideals for \(k\geq 6.\)
In this work he first shows the set of elements of the form \([(x,y,z)^ 2, t]\) generates a nonzero trivial nuclear ideal in \(A_ 4\). For \(k\geq 5\) he next exhibits a function which generates a trivial nuclear ideal in \(A_ k\) but not \(A_{k+1}\). Finally, he gives an example of a nonnuclear ideal in \(A_ k\), where \(k\geq 10\), whose square is a trivial nuclear ideal.
Reviewer: H.F.Smith

MSC:
17D05 Alternative rings
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References:
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