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2013 | OriginalPaper | Chapter

7. Essential Overring Extensions-Beyond the Maximal Ring of Quotients

Authors : Gary F. Birkenmeier, Jae Keol Park, S. Tariq Rizvi

Published in: Extensions of Rings and Modules

Publisher: Springer New York

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Abstract

The focus of this chapter is on right essential overrings of a ring which are not right rings of quotients. Osofsky’s well-known example of a finite ring whose injective hull has no compatible ring structure is considered and generalized. All possible right essential overrings of the ring in Osofsky’s example are discussed. A ring R is constructed with a module essential extension S which is not the injective hull of R. However, S is shown to have one compatible ring structure which is a QF-ring and another compatible ring structure which is not even right FI-extending. Finally, Osofsky compatibility is discussed and a class of rings whose injective hulls have distinct compatible ring structures is studied.

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previous chapter Matrix, Polynomial, and Group Ring Extensions

next chapter Ring and Module Hulls

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Title: Essential Overring Extensions-Beyond the Maximal Ring of Quotients
Authors: Gary F. Birkenmeier
Jae Keol Park
S. Tariq Rizvi
Publisher: Springer New York
Book: Extensions of Rings and Modules
Print ISBN: 978-0-387-92715-2

Electronic ISBN: 978-0-387-92716-9

Copyright Year: 2013
DOI: https://doi.org/10.1007/978-0-387-92716-9_7

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