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E-ideals in baric algebras. (English) Zbl 0868.17023
If p(x)=x n +γ 1 ω(x)x n-1 ++γ n-1 ω(x)x is a train polynomial in a baric algebra (A,ω) over a field F, then E A (1,γ 1 ,,γ n-1 ):=E A (p) is the ideal generated by all p(a) with aA (x n+1 :=x n x). For every (A,ω) and every train polynomial, p(x):E A (p)E A (1,-1). If the baric algebra satisfies (x 2 ) 2 =ω(x) 3 x, then there is a Peirce decomposition A=FeN 1/2 N -1/2 relative to an idempotent e. There are three equivalence classes of train polynomials, leading to either E A (p)=N 1/2 N -1/2 N -1/2 =E A (1,-1) or E A (p)=N 1/2 N -1/2 2 N -1/2 2 or E A (p)=(0).
17D92Genetic algebras
17A30Nonassociative algebras satisfying other identities