Goldie dimensions of quotient modules (Q2748387)

The author defines a right quotient \(\aleph^<\)-dimensional ring \(R\) as one where every direct sum of \(\alpha\) right \(R\)-submodules of any cyclic right \(R\)-module has the property that \(\alpha<\aleph\). It is known that \(R\) is right quotient \(\aleph_0^<\)-dimensional iff every direct sum of injective modules is weakly injective [see \textit{S. K. Jain} and \textit{S. R. López-Permouth}, in: Computational algebra, Lect. Notes Pure Appl. Math. 151, 205-232 (1994; Zbl 0802.16005)]. A right \(R\)-module \(M\) is defined to be weakly very injective (w.v.i.) if for any finitely generated submodule \(\aleph\) of \(E(M)\), the injective hull of \(M\), there exists a triple \(X,D,\psi\) where: \(D\subset_eM\cap N\subset N\subset X\subset E(M)\) and \(X\simeq M\) under an isomorphism \(\psi\colon X\to M\) with \(\psi|_D=\) identity. If in this definition \(D\) is omitted, then \(M\) is called weakly injective. Next, a right \(R\)-module \(M\) is called very tight if for any finitely generated \(\aleph\subset E(M)\), there exists \(D,f\) with \(D\subset_e M\cap\aleph\subset_e\aleph\) and 1-1 homomorphism \(f\colon N\to M\) with \(f|_D=I_D\).NEWLINENEWLINENEWLINEThe main theorem of the paper proves the equivalence of the following statements:NEWLINENEWLINENEWLINETheorem. Let \(\aleph\geq\aleph_0\) be regular. Then for any ring \(R\) the following are equivalent: (1) \(R_R\) is q.\(\aleph^<\)-d. (2) For any \(\Gamma\) and injective modules \(F_\gamma\), \(\gamma\in\Gamma\), \(\prod_{\gamma\in\Gamma}^{<\aleph}F_\gamma\) is weakly very injective. (3) For any \(\Gamma\) and injective modules \(F_\gamma\), \(\gamma\in\Gamma\), \(\prod_{\gamma\in\Gamma}^{<\aleph}F_\gamma\) is very tight. (4) For any \(\Gamma\) and indecomposable injectives \(F_\gamma\), \(\gamma\in\Gamma\), \(\prod_{\gamma\in\Gamma}^{<\aleph}F_\gamma\) is very tight. (5) For any \(\Gamma\) and simple \(S_\gamma\), \(\gamma\in\Gamma\), \(\prod_{\gamma\in\Gamma}^{<\aleph}\widehat S_\gamma\) is very tight.NEWLINENEWLINENEWLINEThe paper concludes with a conjecture and a question conjecture: For every (regular) \(\aleph>\aleph_0\), the following hold: (1) \(R\) is q.\(\aleph^<\)-d. does not imply that for any w.v.i. \(F_\gamma\), \(\gamma\in\Gamma\), \(\prod_{\gamma\in\Gamma}^{<\aleph}F_\gamma\) is w.v.i. (2) If \(R\) has the property that for any injective modules \(F_\gamma\), \(\gamma\in\Gamma\), \(\prod_{\gamma\in\Gamma}^{<\aleph}F_\gamma\) is w.i., then that does not imply that \(R\) is q.\(\aleph^<\)-d. (3) Definition 2.1 and Definition 2.2 cannot be simplified by taking \(D\ll M\cap N\) always to be \(D=M\cap N\). That is, the latter would define different concepts.