Book contents
- Frontmatter
- Contents
- Preface
- Note to the reader
- Terminology, notation and conventions used
- List of special notation
- 0 Generalities on rings and modules
- 1 Principal ideal domains
- 2 Firs, semifirs and the weak algorithm
- 3 Factorization in semifirs
- 4 Rings with a distributive factor lattice
- 5 Modules over firs and semifirs
- 6 Centralizers and subalgebras
- 7 Skew fields of fractions
- Appendix
- Bibliography and author index
- Subject Index
6 - Centralizers and subalgebras
Published online by Cambridge University Press: 22 August 2009
- Frontmatter
- Contents
- Preface
- Note to the reader
- Terminology, notation and conventions used
- List of special notation
- 0 Generalities on rings and modules
- 1 Principal ideal domains
- 2 Firs, semifirs and the weak algorithm
- 3 Factorization in semifirs
- 4 Rings with a distributive factor lattice
- 5 Modules over firs and semifirs
- 6 Centralizers and subalgebras
- 7 Skew fields of fractions
- Appendix
- Bibliography and author index
- Subject Index
Summary
The first topic of this chapter is commutativity in firs. We shall find that any maximal commutative subring of a 2-fir with strong DFL is integrally closed (Corollary 1.2), and the same method allows us to describe the centres of 2-firs as integrally closed rings and make a study of invariant elements in 2-firs and their factors in Sections 6.1 and 6.2. The well-known result that a simple proper homomorphic image of a principal ideal domain is a matrix ring over a skew field is generalized here to atomic 2-firs (Theorem 2.4). In Section 6.3 the centres of principal ideal domains are characterized as Krull domains. Further, the centre of a non-principal fir is shown to be a field in Section 6.4.
Secondly we look at subalgebras and ideals of free algebras in Section 6.6; by way of preparation submonoids of free monoids are treated in section 6.5. A brief excursion into coding theory shows how the Kraft–McMillan inequality can be used to find free subalgebras, and the fir property of free algebras is again derived (Theorem 6.7). Section 6.7 is devoted to a fundamental theorem on free algebras: Bergman's centralizer theorem (Theorem 7.7).
Section 6.8 deals with invariants under automorphisms of free algebras, and Section 6.9 treats the Galois correspondence between automorphism groups and free subalgebras, as described by Kharchenko.
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- Free Ideal Rings and Localization in General Rings , pp. 331 - 409Publisher: Cambridge University PressPrint publication year: 2006