Injective hulls with distinct ring structures.
type: Item
instance of: scholarly article
title: Injective hulls with distinct ring structures.
zbMATH Open document ID: 1170.16001
author: G. F. Birkenmeier
author: Barbara L. Osofsky
author: Jae Keol Park
author: S. Tariq Rizvi
published in: Journal of Pure and Applied Algebra
publication date: 2009-03-10T00:00:00Z
review text: If \(R\) is a ring with \(E=E_R=E(R_R)\) an injective hull and \(\iota\colon R_R\to E_R\) an essential embedding, then the ring structure on \(E(R_R)\) is said to be compatible if the ring multiplication extends the scalar multiplication. It is well known that if \(E(R_R)\) is a rational extension of \(R_R\), then \(E(R_R)\) has a unique compatible ring structure, but \textit{B. L. Osofsky} [Can. Math. Bull. 7, 405-413 (1964; Zbl 0126.06401)] constructed a ring such that no injective hull can possibly have a compatible ring structure. In this paper the question is asked, and answered, whether the existence of a compatible ring structure necessarily implies the uniqueness thereof. A class of right Osofsky compatible rings \(R\) is constructed such that the injective hulls of the rings have more than one ring structure. All of these are isomorphic though, and have the property that they are left Osofsky compatible. The rings used here have \(Q(R)=R\), which is as far as possible from the case where \(Q(R)=E(R_R)\). A final example illustrates the Osofsky compatible ring structures on \(E_R\) using matrix operations.
reviewed by: Frieda Theron
Mathematics Subject Classification ID: 16D50
Mathematics Subject Classification ID: 16S90
Mathematics Subject Classification ID: 16L60
zbMATH DE Number: 5527494
zbMATH Keywords: injective hulls
zbMATH Keywords: compatible ring structures
zbMATH Keywords: Osofsky compatible rings
zbMATH Keywords: ring multiplications
zbMATH Keywords: scalar multiplications
zbMATH Keywords: Morita dualities
zbMATH Keywords: maximal rings of quotients
zbMATH Keywords: quasi-Frobenius rings
MaRDI profile type: Publication
full work available at URL: j.jpaa.2008.09.014
OpenAlex ID: W2001868470
Wikidata QID: Q114155515
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DOI: 10.1016/J.JPAA.2008.09.014
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description: scientific article; zbMATH DE number 5527494