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Essential Overring Extensions-Beyond the Maximal Ring of Quotients

In: Extensions of Rings and Modules

Author

Listed:
  • Gary F. Birkenmeier

    (University of Louisiana at Lafayette, Department of Mathematics)

  • Jae Keol Park

    (Busan National University, Department of Mathematics)

  • S. Tariq Rizvi

    (The Ohio State University at Lima, Department of Mathematics)

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.

Suggested Citation

  • Gary F. Birkenmeier & Jae Keol Park & S. Tariq Rizvi, 2013. "Essential Overring Extensions-Beyond the Maximal Ring of Quotients," Springer Books, in: Extensions of Rings and Modules, edition 127, chapter 0, pages 217-266, Springer.
    • Handle: RePEc:spr:sprchp:978-0-387-92716-9_7
    DOI: 10.1007/978-0-387-92716-9_7
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