Uniform dimension and subdirect products
type: Item
instance of: scholarly article
title: Uniform dimension and subdirect products
zbMATH Open document ID: 0597.16020
author: John Dauns
published in: Pacific Journal of Mathematics
publication date: 1987-01-01T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 16D50
Mathematics Subject Classification ID: 16P60
Mathematics Subject Classification ID: 16D80
Mathematics Subject Classification ID: 16Gxx
Mathematics Subject Classification ID: 16D70
zbMATH DE Number: 3961882
zbMATH Keywords: infinite regular cardinal
zbMATH Keywords: injective hulls
zbMATH Keywords: singular submodules
zbMATH Keywords: direct sum of nonsingular right ideals
zbMATH Keywords: complement submodule
zbMATH Keywords: uniform submodules
zbMATH Keywords: indecomposable injectives
zbMATH Keywords: uniform dimensions
MaRDI profile type: Publication
full work available at URL: pjm.1987.126.1
OpenAlex ID: W2021264392
DOI: 10.2140/PJM.1987.126.1
description: scientific article; zbMATH DE number 3961882