On injective \(L\)-modules. (Q2570950)

The author gives an alternate definition for injective \(L\)-modules as previously given by Zahedi and Ameri, where \(L(\vee,\wedge,1,0)\) is a complete Brouwerian lattice with maximal element 1 and minimal element 0. He proves that a direct sum of \(L\)-modules is injective if and only if each \(L\)-module in the sum is injective. He also proves that if \(J\) is an injective module and \(\mu\) is an injective \(L\)-submodule of \(J\), and if \(0\to\mu\to v\to\eta\to 0\) is a short exact sequence of \(L\)-modules, then \(v\simeq\mu\oplus\eta\), where \(\simeq\) denotes weak isomorphism. Throughout the paper \(R\) is a ring with identity and \(M\) is a module over \(R\). \(L(M)\) denotes the set of all \(L\)-submodules of \(M\). The two main results of the paper are as follows: Let \(J\) be an injective module and \(\mu\in L(J)\) an injective \(L\)-module. If \(0\to J@>f>>B@>g>>C\to 0\) is a short exact sequence of \(R\)-modules and \(v\in L(B)\) and \(\eta\in L(C)\) are such that \(0\to\mu@>f>>v@>g>>\eta\to 0\) is a short exact sequence of \(L\)-modules where \(f(\mu)=v\) on \(f(J)\), then \(v\) is weakly isomorphic to \(\mu\oplus\eta\). Let \(Q_\alpha\), \(\alpha\in I\), be injective \(R\)-modules and \(\mu_\alpha\in L(Q_\alpha)\), \(\alpha\in I\). Then \(\bigoplus_{\alpha\in I}\mu_\alpha\in L(\bigoplus_{\alpha\in I}Q_\alpha)\) is injective if and only if \(\mu_\alpha\) is injective for all \(\alpha\in I\).