The radical in alternative baric algebras
Let \(A\) be an algebra over a field \(F\), and \(\omega : A \rightarrow F\) a nonzero homomorphism. The ordered pair \((A, \omega\)) is called a baric algebra and \(\omega\) its weight. For \(B \subset A\) put \(\text{bar}(B)=\{x \in B : \omega(x)=0\}\). An algebra \(A\) is called an alternative algebra if \(x^2 y = x(xy)\), \(yx^2=(yx)x\), for all \(x, y \in A\). It is proved that in a baric, power associative, algebra over \(F\) of finite dimension, there is an idempotent element of weight 1. In the case when this algebra is alternative the Peirce's decomposition with this idempotent is considered. The nil radical as the maximal nil ideal is defined. It is proved that the bar-radical is the intersection of the nil radical with the square of \(\text{bar}(A)\).
scientific article; zbMATH DE number 1547965