Uniform dimension and subdirect products (Q1078642)

Let \(\sigma\) be an infinite regular cardinal; E denotes injective hulls, Z singular submodules over a ring R. General Theorem. (1)\(\Leftrightarrow (2)\Leftrightarrow (3)\) are equivalent. (1) Any direct sum of nonsingular right ideals of R contains strictly less than \(\sigma\) nonzero summands. (2) For any modules \(W_{\gamma}\), \(\gamma\in \Gamma\), with \(ZW_{\gamma}=0\) and arbitrary \(\Gamma\) ; the submodule \(\prod^{\sigma}E(W_{\gamma})\equiv \{x=(x_{\gamma})\in \prod E(W_{\gamma})|\) \(| \sup port\) \(x| <\sigma \}\) is injective. (3) For any \(W_{\gamma}\), \(\gamma\in \Gamma\), \(ZW_{\gamma}=0\); \(\prod^{\sigma}W_{\gamma}\leq \prod W_{\gamma}\) is a complement submodule. Corollary. Every ring R satisfies (1), (2), and (3) for a unique smallest infinite regular cardinal \(\sigma =\sigma (R)\). Corollaries and applications are given of the following theorem. For any module M with \(ZM=0\), \(E(M)=C\oplus D\); C contains no uniform submodules. Both C and \(D=\prod \{E(D_{\tau})|\) \(\tau\in \Xi \}\) are unique, as well as each \(D_{\tau}\), which is a direct sum of isomorphic indecomposable injectives all of the same type \(\tau\). The cardinal number of such summands is the \(\tau\)-dimension of M. More general uniform dimensions are constructed for arbitrary modules.