Zassenhaus algebras.
If \(R\) is a ring and \(\mathcal F\) is a family of left ideals of \(R\), it is very useful to consider the subring \(\mathrm{End}(R,\mathcal F)\) of \(\mathrm{End}(R^+)\) which consists in all endomorphisms \(\varphi\) such that \(\varphi(X)\subseteq X\) for all \(X\in\mathcal F\). If \(\mathcal F\) consists of all left ideals of \(R\), then this subring is denoted by \(\widehat R\), and it consists of all endomorphisms of the Abelian group \(R^+\) which leave all left ideals invariant. These subrings are very useful in the constructions of modules (Abelian groups) with the endomorphism ring isomorphic to \(R\), [e.g. \textit{H. Zassenhaus} in J. Lond. Math. Soc. 42, 180-182 (1967; Zbl 0145.26405)]. The main aim of the present paper is to study \(\widehat R_K\) for the case \(R\) is a \(K\)-algebra over the field \(K\). There are many interesting results in this paper, and we will exemplify these with few examples. It is proved that if \(R\neq K\) is a finite dimensional local \(K\)-algebra and \(K\) is infinite then \(R\neq\widehat R_K\), i.e. \(R\) is not a Zassenhaus algebra, but \(\widehat{\widehat R}_K=\widehat R_K\), i.e. \(\widehat R_K\) is a Zassenhaus algebra (Theorem 3). If \(K\) is an infinite field and \(Q\) is a finite quiver, then the path algebra \(KQ\) is a Zassenhaus algebra. The same conclusion is valid for arbitrary fields \(K\) if we suppose that \(Q\) has not loops or it is acyclic (Theorem 5). It is also proved that the operator \(\widehat -\) acts in many important cases as a closure operator (Theorem 10).
scientific article; zbMATH DE number 5996725