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
9 - 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 introduces the notion of a ring, more specifically, a commutative ring with unity. The theory of rings provides a useful conceptual framework for reasoning about a wide class of interesting algebraic structures. Intuitively speaking, a ring is an algebraic structure with addition and multiplication operations that behave like we expect addition and multiplication should. While there is a lot of terminology associated with rings, the basic ideas are fairly simple.
Definitions, basic properties, and examples
Definition 9.1. A commutative ring with unityis a set R together with addition and multiplication operations on R, such that:
(i) the set R under addition forms an abelian group, and we denote the additive identity by 0R;
(ii) multiplication is associative; that is, for all a, b, c ∈ R, we have a(bc) = (ab)c;
(iii) multiplication distributes over addition; that is, for all a, b, c ∈ R, we have a(b + c) = ab + ac and (b + c)a = ba + ca;
(iv) there exists a multiplicative identity; that is, there exists an element 1R ∈ R, such that 1R · a = a = a · 1R for all a ∈ R;
(v) multiplication is commutative; that is, for all a, b ∈ R, we have ab = ba.
There are other, more general (and less convenient) types of rings–one can drop properties (iv) and (v), and still have what is called a ring.
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- Information
- A Computational Introduction to Number Theory and Algebra , pp. 211 - 243Publisher: Cambridge University PressPrint publication year: 2005