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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
Full Text:
References:
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