This paper is motivated by an interest in the relation between the supremum of the projective lengths of injective left \(R\)-modules (\(\text{spli\,}R\)) and the supremum of the injective lengths of projective left \(R\)-modules (\(\text{silp\,}R\)), as introduced by T. V. Gedrich and K. W. Gruenberg [Topology Appl. 25, 203-223 (1987; Zbl 0656.16013)]. The corresponding invariants for right \(R\)-modules, \(\text{spli\,}R^{op}\) and \(\text{silp\,}R^{op}\) are also considered. The notion of the supremum of the flat lengths of injective left \(R\)-modules (\(\text{sfli\,}R\)), strict Mittag-Leffler modules and a certain natural transformation \(\Phi\), which involves both Tor and Ext groups play an essential role in the proofs of results.
Strict Mittag-Leffler modules are first examined. A property of their representation as a direct limit of finitely presented modules is related to the natural transformation \(\Phi\). Using the invariant \(\text{sfli\,}R\), it is shown that, for a left and right \(\aleph_0\)-Noetherian ring \(R\), if the invariants \(\text{spli\,}R\) and \(\text{spli\,}R^{op}\) are finite, then they are either equal or they differ by 1. This result is used to bound the finitistic dimension of a left perfect or a countable ring by the injective dimension of the left regular module.
In the case of a left \(\aleph_0\)-Noetherian ring \(R\), it is shown that \(\text{silp\,}R=\text{id}_RR^{(\mathbb N)}\). If \(R\) is isomorphic to its opposite \(R^{op}\) and \(\text{id}_R<\infty\), then we have the equality \(\text{spli\,}R=\text{silp\,}R\).
The final section considers the special case where \(R\) is the integral group ring \(\mathbb ZG\) of a group \(G\). If \(G\) is a countable group, the following results are proved:(1) If \(\text{cd\,}G=n\), then \(H^n(G,\mathbb ZG)\neq 0\). (2) If \(\text{silp\,}\mathbb ZG\) is finite, then \(\text{silp\,}\mathbb ZG=\text{id}_{\mathbb ZG}\mathbb ZG\). The generalized cohomological dimension \(\underline{\text{cd}}\,G\) of a group has been defined by B. M. Ikenaga [J. Algebra 87, 422-457 (1984; Zbl 0536.20032)] as the supremum of those integers \(n\), for which there exists a \(\mathbb Z\)-free \(\mathbb ZG\)-module \(M\) and a projective \(\mathbb ZG\)-module \(P\), such that \(\text{Ext}_{\mathbb ZG}^n(M,P)\neq 0\). The generalized homological dimension \(\underline{\text{hd}}\,G\) of a group \(G\) is defined as the supremum of those integers \(n\) for which there exists a \(\mathbb Z\)-free \(\mathbb ZG\)-module \(M\) and an injective \(\mathbb ZG\)-module \(I\), such that \(\text{Tor}_n^{\mathbb ZG}(I,M)\neq 0\). For any group, it is shown that the generalized homological dimension of \(G\) is less than or equal to the cohomological dimension of \(G\). A class of examples of groups \(G\) is given for which the inequality \(\text{sfli\,}\mathbb ZG\leq\text{spli\,}\mathbb ZG\) is strict.