Strongly regular elements of Noetherian rings
Let R be a right Noetherian ring. As usual, for any ideal I of \(R,\) C'(I)\(=\{c\in R:\) \(r\in R\), cr\(\in I\) implies \(r\in I\}\), \('C(I)=\{c\in R:\) \(r\in R\), rc\(\in I\) implies \(r\in I\}\) and \(C(I)=C'(I)\cap 'C(I)\). In this paper interest centres on \(S'(0)=\cap \quad C'(I)\) where the intersection is taken over all right annihilator ideals I, \('S(0)=\cap \quad 'C(J)\) where the intersection is taken over all left annihilator ideals J, and \(S(0)=S'(0)\cap 'S(0).\) The elements of C(0) and S(0) are called regular and strongly regular, respectively. Clearly \(S'(0)\subseteq C'(0),\quad 'S(0)\subseteq 'C(0)\) and \(S(0)\subseteq C(0).\) The set S(0) is of interest because it contains any right Ore set of regular elements of R and hence R has a (classical) right quotient ring only if \(C(0)=S(0)\). However an example is given of a Noetherian PI- ring R of Krull dimension 1 which satisfies \(C(0)=S(0)\) but which has neither a left nor a right quotient ring. Investigation of the set S(0) for an arbitrary right Noetherian ring R takes different forms. For example, various necessary conditions on R are given which imply \(C(0)=S(0).\) Next for a Noetherian ring R, \(S(0)=S'(0)='S(0)=\cap\) \(C'(M)=\cap\) C(P), where the first intersection is taken over all middle annihilators M of R and the second over all prime middle annihilators P of R, and S(0) is saturated in the sense that st\(\in S(0)\) implies \(s\in S(0)\) and \(t\in S(0)\). This raises the question whether a Noetherian ring has a finite collection of prime middle annihilators \(P_ i\) (1\(\leq i\leq n)\) such that \(S(0)=\cap^{n}_{i=1}C(P_ i).\) Although this question is not settled in general, various cases are given where it is true, for example homomorphic images of enveloping algebras of finite dimensional Lie algebras or of group algebras of polycyclic-by-finite groups, FBN-rings, rings which are finitely generated modules over their Noetherian centres, and Noetherian rings R such that S(0) is a right Ore set and the right quotient ring \(R_{S(0)}\) is semi-local with only finitely many maximal middle annihilators whose intersection is the Jacobson radical of \(R_{S(0)}\). Under localization, S(0) behaves well; in fact if \(\Omega\) is a right Ore set of regular elements of R and Q the right quotient ring of R with respect to \(\Omega\) then, for Q, \(S_ Q(0)=\{sc^{-1}:\) \(s\in S(0),\) \(c\in \Omega\}.\)
enveloping algebras of finite dimensional Lie algebras
group algebras of polycyclic-by-finite groups
scientific article; zbMATH DE number 3877370
scientific article