On the Krull dimension of endo-bounded modules. (Q2873171)

Since it appeared in the early 1970s, the well-known theorem about the equality of the classical and module-theoretic (in the sense of Gabriel and Rentschler) Krull dimensions of a fully bounded Noetherian ring has been generalized in various ways. The paper under review presents another result of this kind.NEWLINENEWLINE Let \(R\) be a ring with unit element, let \(M\) be a unitary right \(R\)-module, and let \(\mathcal L_2(M)\) denote the set of fully invariant submodules of \(M\). Then \(M\) is called \(\mathcal L_2\)-Noetherian if the members of \(\mathcal L_2(M)\) are finitely generated. Call \(P\in\mathcal L_2(M)\) to be \(\mathcal L_2\)-prime if \(\Hom_R(M,W_1)W_2\subseteq P\) implies that \(W_1\subseteq P\) or \(W_2\subseteq P\), whenever \(W_i\in\mathcal L_2(M)\), \(i=1,2\), and denote the set of \(\mathcal L_2\)-prime submodules of \(M\) by \(\mathrm{Spec}_2(M)\). If every essential submodule of \(M\) contains a fully invariant essential submodule, then \(M\) is called endo-bounded, and \(M\) is fully endo-bounded if \(M/P\) is endo-bounded as a module over \(R/\mathrm{ann}_R(M/P)\) for every \(P\in\mathrm{Spec}_2(M)\). The \(\mathcal L_2\)-classical Krull dimension \(\mathcal L_2\mathrm{-dim}(M)\) of \(M\) is the smallest ordinal \(\alpha\) such that \(X_\alpha=\mathrm{Spec}_2(M)\), where \(X_{-1}=\emptyset\) and for an ordinal \(\gamma>-1\), \(X_\gamma=\{P\in\mathrm{Spec}_2(M)\mid P\subset Q\in\mathrm{Spec}_2(M)\Rightarrow Q\in X_\beta\text{ for some }\beta<\alpha\}\).NEWLINENEWLINE One main result of the article asserts that if \(M_R\) is quasi-projective, \(\mathcal L_2\)-Noetherian, fully endo-bounded and has Krull-dimension, then \(\mathrm{K.dim}(M)\leq\mathcal L_2\mathrm{-dim}(M)\). The other main result states that the reverse inequality holds if \(M_R\) is \(\Sigma\)-projective, \(\mathcal L_2\)-Noetherian, has Krull dimension, and satisfies the condition that \(\mathcal L_2\)-prime factors of \(M\) are essentially compressible.