Book contents
- Frontmatter
- Contents
- Preface
- Preliminaries
- 1 Basic properties of the integers
- 2 Congruences
- 3 Computing with large integers
- 4 Euclid's algorithm
- 5 The distribution of primes
- 6 Finite and discrete probability distributions
- 7 Probabilistic algorithms
- 8 Abelian groups
- 9 Rings
- 10 Probabilistic primality testing
- 11 Finding generators and discrete logarithms in
- 12 Quadratic residues and quadratic reciprocity
- 13 Computational problems related to quadratic residues
- 14 Modules and vector spaces
- 15 Matrices
- 16 Subexponential-time discrete logarithms and factoring
- 17 More rings
- 18 Polynomial arithmetic and applications
- 19 Linearly generated sequences and applications
- 20 Finite fields
- 21 Algorithms for finite fields
- 22 Deterministic primality testing
- Appendix: Some useful facts
- Bibliography
- Index of notation
- Index
17 - More rings
Published online by Cambridge University Press: 05 June 2012
- Frontmatter
- Contents
- Preface
- Preliminaries
- 1 Basic properties of the integers
- 2 Congruences
- 3 Computing with large integers
- 4 Euclid's algorithm
- 5 The distribution of primes
- 6 Finite and discrete probability distributions
- 7 Probabilistic algorithms
- 8 Abelian groups
- 9 Rings
- 10 Probabilistic primality testing
- 11 Finding generators and discrete logarithms in
- 12 Quadratic residues and quadratic reciprocity
- 13 Computational problems related to quadratic residues
- 14 Modules and vector spaces
- 15 Matrices
- 16 Subexponential-time discrete logarithms and factoring
- 17 More rings
- 18 Polynomial arithmetic and applications
- 19 Linearly generated sequences and applications
- 20 Finite fields
- 21 Algorithms for finite fields
- 22 Deterministic primality testing
- Appendix: Some useful facts
- Bibliography
- Index of notation
- Index
Summary
This chapter develops a number of other concepts concerning rings. These concepts will play important roles later in the text, and we prefer to discuss them now, so as to avoid too many interruptions of the flow of subsequent discussions.
Algebras
Let R be a ring. An R-algebra (or algebra overR) is a ring E, together with a ring homomorphism τ: R → E. Usually, the map τ will be clear from context, as in the following examples.
Example 17.1. If E is a ring that contains R as a subring, then E is an R-algebra, where the associated map τ: R → E is just the inclusion map.
Example 17.2. Let E1, …, En be R-algebras, with associated maps τi: R → Ei, for i = 1, …, n. Then the direct product ring E:= E1 × … × En is naturally viewed as an R-algebra, via the map τ that sends a ∈ R, to (τ1(a), …, τn(a)) ∈ E.
Example 17.3. Let E be an R-algebra, with associated map τ: R → E, and let I be an ideal of E. Consider the quotient ring E/I. If ρ is the natural map from E onto E/I, then the homomorphism ρ ∘ τ makes E/I into an R-algebra, called the quotient algebra ofEmoduloI.
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- A Computational Introduction to Number Theory and Algebra , pp. 359 - 397Publisher: Cambridge University PressPrint publication year: 2005