Hulls of Ring Extensions

Gary F. Birkenmeier, Jae Keol Park and S. Tariq Rizvi
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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

Chapter Chapter 9 in Extensions of Rings and Modules, 2013, pp 327-354 from Springer

Abstract: Abstract In this chapter, the theory of ring hulls is used to determine when various ring extensions are in the classes of interest (e.g., right (FI-) extending, (quasi-) Baer, etc.) or when certain subrings (e.g., the fixed ring) are in these classes. Section 9.1 begins with a characterization of a right extending ring whose maximal right ring of quotients is the 2×2 matrix ring over a division ring. This result eventually leads to a characterization of all right rings of quotients of a 2×2 upper triangular matrix ring over a commutative PID which are right extending, Baer, right Rickart, or right semihereditary. Skew group rings and fixed rings are considered in Sect. 9.2. The main results of this section concern semiprime rings with a group of X-outer ring automorphisms which have their skew group ring and/or fixed ring being quasi-Baer. In the final section, various matrix ring extensions (both finite and infinite) and monoid ring extensions of a ring hull are compared to the corresponding ring hull of the matrix or monoid ring extension. Moreover, for a semiprime ring R which is Morita equivalent to a ring S, then their quasi-Baer ring hulls are also Morita equivalent.

Keywords: Ring extension; Field of fractions; Commutative PID; Skew group ring; X-outer ring automorphism; Fixed ring; G-quasi-Baer; Infinite matrix extensions; Monoid ring; Morita equivalent (search for similar items in EconPapers)
Date: 2013
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DOI: 10.1007/978-0-387-92716-9_9

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