Injectivity conditions over a formal triangular matrix ring
If \(A\) and \(B\) are (associative with \(1\neq 0\)) rings and \(M\) is a left-\(B\), right-\(A\) (unital) bimodule, let \(T\) be the set of formal triangular matrices of the form \(\left(\begin{smallmatrix} a &0\\ m&b\end{smallmatrix}\right)\) , with \(a\in A\), \(b\in B\) and \(m\in M\). Then \(T\) forms a ring under component-wise addition and multiplication given by a rule similar to matrix multiplication. It is well-known that \(\text{Mod-}T\) is equivalent to the category \(\Omega\) of triples \((X,Y)_f\) with \(X\in\text{Mod-}A\), \(Y\in\text{Mod-}B\) and \(f\colon Y\otimes_RM\to X\) a map in \(\text{Mod-}A\). If the right \(T\)-module \(V\) corresponds to \((X,Y)_f\), then \(\widetilde f\colon Y\to\Hom(M,X)\) given by \(\widetilde f(m)=f(y\otimes m)\) for \(y\in Y\), \(m\in M\), is a \(B\)-homomorphism. Let \(V_1\) and \(V_2\) be right \(T\)-modules with \((X_1,Y_1)_{f_1}\) and \((X_2,Y_2)_{f_2}\) the corresponding triples. It is shown that if \(X_1\) is \(X_2\)-injective in \(\text{Mod-}A\) and \(\widetilde f_1\) is an isomorphism in \(\text{Mod-}B\), then \(V_1\) is \(V_2\)-injective in \(\text{Mod-}T\). If \(V\) and \(W\) are \(R\)-modules, the simple reject \(\text{s-rej}(W,V)\) is defined to be the intersection of all submodules \(\ker\phi\) where \(\phi\colon W\to V\) is an \(R\)-map with \(\text{Im }\phi\cap\text{Soc }V\neq 0\); if no such \(\phi\) exists, \(\text{s-rej}(W,V)\) is defined to be \(W\). The simple trace \(\text{s-tr}(W,V)\) is the sum of all submodules \(\text{Im }\phi\) where \(\phi\colon W'\to V\), \(W'\) a submodule of \(W\) and \(\text{Im }\phi\cap\text{Soc }V\neq 0\); if no such \(\phi\) exists, set \(\text{s-tr}(W,V)=0\). It is proved that \(V_1\) is simple \(V_2\)-injective in \(\text{Mod-}T\) if and only if \(X_1\) is simple \(X_2\)-injective in \(\text{Mod-}A\) and \(Y_1\) is simple \(Y_2\)-injective in \(\text{Mod-}B\), provided some conditions on \(\text{s-rej}(X_1,X_2)\) and \(\text{s-tr}(Y_2,Y_1)\) hold. The injective cogenerators of \(\text{Mod-}T\) are also studied in terms of properties of the representation triples. Various examples complement the theory.
scientific article; zbMATH DE number 1773424
scientific article