An equivalence criterion for the generalized injectivity of modules with respect to algebraic classes of homomorphisms (Q2816940)

The authors are inspired by some results of Bumby, such as the following: Two modules which are isomorphic to submodules of each other have isomorphic injective hulls. The authors then liken the situation to Cantor-Bernstein-Schröder theorem (CBST) for cardinality of sets, which they emulate in the context of (module) categories. For that purpose \(\mathcal H\) denotes any family of monic morphisms and an object \(Q\) is defined to be \(\mathcal H\)-injective, if, for every pair of objects \(A, B\) and every morphism \(\alpha\) in the set hom\(_\mathcal H(A,B)\) of homomorphisms in \(\mathcal H\) and every morphism \(\phi:A\longrightarrow Q\), there exists a morphism \(\psi:B\longrightarrow Q\) such that \(\psi\alpha=\phi\). This is a generalization of the notion of injectivity, but module categories often contain modules that are \(\mathcal H\)-injective, but not injective. Nonetheless, the usual set of properties of injective modules readily carries over into this generally wider class. An object \(A\) is an \(\mathcal H\)-subobject of an object \(B\) if the set hom\(_\mathcal H(A,B)\) is non-empty and \(A\) and \(B\) are \(\mathcal H\)-equivalent if they are \(\mathcal H\)-subobjects of each other. A morphism \(\alpha\in\mathcal H\) is \(\mathcal H\)-essential, if \(\beta\alpha\in \mathcal H\) implies \(\beta\in\mathcal H\), for every \(\beta\). An \(\mathcal H\)-injective hull is then defined appropriately via essential submodules. To make this structure yield a CBST type result, the authors define, for every left \(R\)-module \(A\), the set \(\mathcal H_A\) to be algebraic, if it includes all isomorphisms, if it is closed under composition, direct summands of \(A\), direct sums of \(\mathcal H\)-essential morphisms, unions of countable ascending chains and that every module has an \(\mathcal H\)-injective hull. The main result is as follows: Let \(\mathcal H\) be an algebraic class. Two \(\mathcal H\)-injective objects are isomorphic when they are \(\mathcal H\)-equivalent.