The pure-injective and RD-injective hulls of a ring
type: Item
instance of: scholarly article
title: The pure-injective and RD-injective hulls of a ring
zbMATH Open document ID: 0815.16008
author: G. E. Puninskij
published in: Mathematical Notes
publication date: 1994-07-21T00:00:00Z
review text: \textit{A. Facchini} proved [Q. J. Math., Oxf. II. Ser. 39, 307-321 (1988; Zbl 0668.13012)] that if \(R\) is a commutative ring and if the pure- injective hull of \(R\) (as a module over itself) is indecomposable then \(R\) is local and he asked if the converse is true. The author shows that, in fact, if \(R\) is any ring then its pure-injective hull as a (right, say) module over itself is indecomposable if and only if \(R\) is local. Indeed, \textit{I. Herzog} has shown more generally [Contemp. Math. 130, 153-165 (1992; Zbl 0797.16019)] that if \(M\) is a finitely presented module (over any ring) then the pure-injective hull of \(M\) is indecomposable if and only if \(M\) has local endomorphism ring. An embedding of the module \(M\) into the module \(N\) is said to be RD-pure if, for every element \(r\) of the ring, \(Mr=M \cap Nr\). There is a corresponding notion of RD-injective and [see \textit{A. Facchini}, J. Algebra 110, 380-406 (1987; Zbl 0629.13008)] RD-injective hull. The author provides a criterion for the (right) RD-injective hull of a ring to be indecomposable. He uses this to show that if \(R\) has indecomposable (left or right) RD-injective hull then \(R\) is local and he shows that the converse is false by giving an example of a (local artinian) ring whose right RD-injective hull is indecomposable but whose left RD-injective hull is not. He also notes that the ring \(k[x,y : x^ 2=y^ 2=xy=0]\), where \(k\) is a field, is pure-injective but not RD- injective as a module over itself.
Mathematics Subject Classification ID: 16L30
Mathematics Subject Classification ID: 16D50
Mathematics Subject Classification ID: 16B70
Mathematics Subject Classification ID: 03C60
zbMATH DE Number: 578713
zbMATH Keywords: filtered modules
zbMATH Keywords: pp formulas
zbMATH Keywords: indecomposable module
zbMATH Keywords: local artinian ring
zbMATH Keywords: pure-injective hull
zbMATH Keywords: finitely presented module
zbMATH Keywords: local endomorphism ring
zbMATH Keywords: embedding
zbMATH Keywords: RD-injective hull
reviewed by: Mike Prest
MaRDI profile type: Publication
Recommended article: On Compatible Ring Structures of the Injective Hull of a Ring
Recommended article: Injective hulls of φ-trivial extensions of rings
Recommended article: PROPERTIES OF INJECTIVE HULLS OF A RING HAVING A COMPATIBLE RING STRUCTURE
Recommended article: Rings whose injective hulls are cogenerators
Recommended article: On the rings whose injective hulls are flat
Recommended article: Rings with cyclic injective hulls
Recommended article: Pure-injective hulls of modules over valuation rings
Recommended article: Q4763119
Recommended article: On p-injective rings
Recommended article: On the Pure-injectivity Profile of a Ring
description: scientific article; zbMATH DE number 578713