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$E$-ideals in baric algebras. (English) Zbl 0868.17023
If $p(x)= x^n+ \gamma_1\omega(x) x^{n-1}+\cdots+ \gamma_{n- 1}\omega(x)x$ is a train polynomial in a baric algebra $(A,\omega)$ over a field $F$, then $E_A(1, \gamma_1,\dots, \gamma_{n-1}):= E_A(p)$ is the ideal generated by all $p(a)$ with $a\in A$ $(x^{n+1}:= x^nx)$. For every $(A,\omega)$ and every train polynomial, $p(x): E_A(p)\subseteq E_A(1,-1)$. If the baric algebra satisfies $(x^2)^2= \omega(x)^3x$, then there is a Peirce decomposition $A=Fe\oplus N_{1/2}\oplus N_{-1/2}$ relative to an idempotent $e$. There are three equivalence classes of train polynomials, leading to either $E_A(p)= N_{1/2}N_{-1/2}\oplus N_{-1/2}= E_A(1,-1)$ or $E_A(p)= N_{1/2} N^2_{-1/2}\oplus N^2_{-1/2}$ or $E_A(p)=(0)$.
##### MSC:
 17D92 Genetic algebras 17A30 Nonassociative algebras satisfying other identities