On rings whose number of centralizers of ideals is finite (Q1100257)

Define \(n(R)\) to be the cardinal number of the set of centralizers of nonzero ideals of R. For subsets S, T of R, let (S,T) be the ideal generated by [S,T]; and call any such ideal a commutator ideal. Call R an (N)-ring if no commutator ideal contains a nonzero nilpotent ideal of R. The authors first prove that \(n(R)=1\) if and only if R is commutative or prime, and they then characterize as irredundant subdirect sums the (N)- rings with n(R) finite. For arbitrary positive integers k, they construct rings R with \(n(R)=k\). Finally, they show that if R is the ring of upper- triangular \(k\times k\) matrices over a commutative domain with 1, then \(n(R)=k^ 2-2k+2.\) \{There are numerous typographical errors, some of which obscure the meaning.\}