The lattice structure of preradicals. III: Operators.
Let \(R\) be a ring with identity and denote by \(R\)-pr the class of all the preradicals over \(R\). The authors continue the study of \(R\)-pr. In a previous paper [part II, J. Algebra Appl. 1, No. 2, 201-214 (2002; Zbl 1034.16035)], they defined several preradicals associated to a given preradical. Using these preradicals, they introduced several partitions of \(R\)-pr. It were also shown applications that classify rings. In this paper, they consider several operators acting on \(R\)-pr, for instance the pseudocomplement, the annihilator and the totalizer, as well as some relations among them. A preradical \(\sigma\) is called extreme if \(\sigma(ES)=0\) or \(\sigma(ES)=ES\) for each simple \(R\)-module \(S\), where \(ES\) means the injective hull of \(S\) and the collection of all extreme preradicals over \(R\) is denoted by \(R\)-prex. Then it is shown that \(R\) is a left V-ring if and only if \(R\)-pr=\(R\)-prex. -- Let \(S\) and \(T\) be simple \(R\)-modules. Recall that \(S\) and \(T\) are homologically connected if there exists a finite sequence of simple \(R\)-modules \(S=S_1,S_2,\dots,S_n=T\) such that \(\Hom_R(ES_i,ES_{i+1})\neq 0\) for all \(1\leq i<n\). Then it is also shown that \(R\) is a finite direct sum of \(ES\) with \(S\) simple and for each pair of simple \(R\)-modules \(S,T\), \(S\) and \(T\) are homologically connected if and only if \(R\)-prex=\(\{0,1\}\). -- A ring that is a finite direct sum of \(ES\) with \(S\) simple is also characterized, but it is too technical to state here.
scientific article; zbMATH DE number 2093459
scientific article