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Baer, Rickart, and Quasi-Baer Rings

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

This chapter is devoted to properties of rings for which certain annihilators are direct summands. Such classes of rings include those of Baer rings, right Rickart rings, quasi-Baer rings, and right p.q.-Baer rings. The results and the material presented in this chapter will be instrumental in developing the subject of our study in later chapters. It is shown that the Baer and the Rickart properties of rings do not transfer to the rings of matrices or to the polynomial ring extensions, while the quasi-Baer and the p.q.-Baer properties of rings do so. The notions of Baer and Rickart rings are compared and contrasted in Sect. 3.1 and the notions of quasi-Baer and principally quasi-Baer rings in Sect. 3.2, respectively. A result of Chatters and Khuri shows that there are strong bonds between the extending and the Baer properties of rings. We shall also see some instances where the two notions coincide. It is shown that there are close connections between the FI-extending and the quasi-Baer properties for rings. One of the motivations for the study of the quasi-Baer and p.q.-Baer rings is the fact that they behave better with respect to various extensions than the Baer and Rickart rings. For example, as shown in this chapter, each of the quasi-Baer and the p.q.-Baer properties is Morita invariant. This useful behavior will be effectively applied in later chapters. The results on the transference (or the lack of transference) of the properties presented in this chapter to matrix and polynomial ring extensions are intended to motivate further investigations on when these properties transfer to various other extensions.

Suggested Citation

  • Gary F. Birkenmeier & Jae Keol Park & S. Tariq Rizvi, 2013. "Baer, Rickart, and Quasi-Baer Rings," Springer Books, in: Extensions of Rings and Modules, edition 127, chapter 0, pages 61-92, Springer.
    • Handle: RePEc:spr:sprchp:978-0-387-92716-9_3
    DOI: 10.1007/978-0-387-92716-9_3
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