Rings of endomorphisms and distributivity
The ring of endomorphisms \(R= \text{End}(M_A)\) of a distributive module \(M_A\) is studied. If \(T\) is the set of all nilpotent elements of \(R\), then \(T\) is a nil-subring of \(R\) and \(T\subseteq F(M)\cap G(M)\cap J(R)\), where \(F(M)= \{f\in R\mid \text{Im } f\) is small in \(M_A\}\), \(G(M)= \{f\in R\mid \text{Ker } f\) is essential in \(M_A\}\), and \(J(R)\) is the Jacobson radical of \(R\). Moreover, the rings \(R/F(M)\), \(R/G(M)\) and \(R/(F(M)\cap G(M))\) are reduced (without nonzero nilpotent elements). In this case, the prime radical of \(R\) is the greatest right nil-ideal and greatest left nil-ideal of \(R\). Some characterizations of a distributive module \(M_A\) are obtained with the help of the ring \(R= \text{End}(M_A)\). For example, \(M_A\) is distributive if the left \(R\)-module \(\text{Hom}_A(M, E)\) is distributive for every quasi-injective module \(E_A\) (it is sufficient to consider the minimal injective cogenerating module \(E_A\) of \(\text{Mod-}A\)). If \(M_A\) is a distributive quasi-injective module, then \(R= \text{End}(M_A)\) is a left distributive Bezout ring and the ring \(R/J(R)\) is strictly regular.
scientific article; zbMATH DE number 830746