Peirce-evanescent baric identities (Q2201063)

In an algebra \(A\) an element \(e\in A\) is an \textit{idempotent} if \(e^2=e\). If \(A\) is associative, then \(e^2x-ex=0\) for every \(x\in A\) and then the left multiplication \(L_e\) is annihilated by the polynomial \(X^2-X\). If \(A\) is not associative, then \(X^2-X\) need not annihilate \(L_e\) but, if we assume that \(A\) satisfies some other identity, it is sometimes possible to find the annihilator of \(L_e\), which is called the \textit{Peirce polynomial of \(A\)} and use it to obtain the so-called \textit{Peirce decomoposition} of the algebra \(A\). However, for some classes of algebras the Peirce polynomial turns out to be the null polynomial and, hence, it does not lead to the Peirce decomposition of \(A\). In such a setting, the author calls the identity satisfied by the algebra \textit{Peirce-evanescent}. In this paper, the author focuses on Peirce-evanescent identities in the context of baric algebras; i.e., \(K\)-algebras \(A\) endowed with an algebra homomorphism \(w:A\rightarrow K\). In this setting, procedures for constructing such homogeneous and non homogeneous identities are given. Many particular examples are worked out, obtaining a little more than 250 evanescent identities.