Relative FBN rings and the second layer condition (Q1295643)

A prime ideal \(P\) of a right Noetherian ring \(R\) satisfies the right second layer condition if the ``second layer'' of the injective hull \(E=E_R(R/P)\) of the right \(R\)-module \(R/P\) is ``tame'' -- that is, every prime submodule of \(E/l\text{-ann}_E(P)\) is torsion-free. This condition was introduced 20 years ago by \textit{A. V. Jategaonkar} [Localization in Noetherian Rings. Lond. Math. Soc. Lect. Note Ser. 98, Cambridge University Press (1986; Zbl 0589.16014)] and the reviewer [\textit{K. A. Brown}, J. Algebra 69, 247--260 (1981; Zbl 0471.16012)], and has been much studied since then. The ring \(R\) satisfies the right second layer condition if all its prime ideals do. In this paper the authors compare and discuss several variants of this definition which have appeared in the literature. In their main result (Theorem 6.2) they provide various equivalent formulations of the condition, in particular giving several such formulations not involving any explicit mention of injective hulls. In doing so they define a new dimension for a module \(M\), the tame dimension, defined as the deviation of the set of submodules \(T\) of \(M\) with \(M/T\) tame, which may have independent applications.