1.28: A Rings and Groups

The material in this appendix is optional reading. However, for the sake of completeness we state here the definition of a ring and the definition of a group. If you are interested in learning more you might take the course Elementary Abstract Algebra. Having had this course should make it a little easier to understand the ideas in abstract algebra and vice versa.

For more details you may download the free book Elementary Abstract Algebra from my homepage:

 http://www.math.usf.edu/~eclark 

Alternatively, look in almost any book whose title contains the words Abstract Algebra or Modern Algebra. Look for one with Introductory or Elementary in the title.

Definition \(\PageIndex{1}\): Ring

A ring is an ordered triple \((R, + ,\cdot)\) where \(R\) is a set and \(+\) and \(\cdot\) are binary operations on \(R\) satisfying the following properties:

A1 \(a + (b+c) = (a+b)+c\) for all \(a\), \(b\), \(c\) in \(R\).

A2 \(a+b=b+a\) for all \(a\), \(b\) in \(R\).

A3 There is an element \(0 \in R\) satisfying \(a+0=a\) for all \(a\) in \(R\).

A4 For every \(a \in R\) there is an element \(b \in R\) such that \(a+b=0\).

M1 \(a \cdot (b \cdot c) = ( a \cdot b ) \cdot c\) for all \(a\), \(b\), \(c\) in \(R\).

D1 \(a \cdot (b+c) = a \cdot b + a \cdot c\) for all \(a\), \(b\), \(c\) in \(R\).

D2 \((b+c) \cdot a = b \cdot a + c \cdot a\) for all \(a\), \(b\), \(c\) in \(R\).

Thus, to describe a ring one must specify three things:

1. a set,
2. a binary operation on the set called multiplication,
3. a binary operation on the set called addition.

Then, one must verify that the properties above are satisfied.

Example \(\PageIndex{1}\)

Here are some examples of rings. The two binary operations \(+\) and \(\cdot\) are in each case the ones that you are familiar with.

1. \((\mathbb{R},+, \cdot)\)–the ring of real numbers.
2. \((\mathbb{Q},+, \cdot)\)–the ring of rational numbers.
3. \((\mathbb{Z},+, \cdot)\)–the ring of integers.
4. \((\mathbb{Z}_n,+, \cdot)\)–the ring of integers modulo \(n\).
5. \((M_n(\mathbb{R}),+, \cdot)\)–the ring of all \(n \times n\) matrices over \(\mathbb{R}\).

Definition \(\PageIndex{2}\): Group

A group is an ordered pair \((G,*)\) where \(G\) is a set and \(*\) is a binary operation on \(G\) satisfying the following properties

1. \(x*(y*z) = (x*y)*z\) for all \(x\), \(y\), \(z\) in \(G\).
2. There is an element \(e \in G\) satisfying \(e*x=x\) and \(x*e=x\) for all \(x\) in \(G\).
3. For each element \(x\) in \(G\) there is an element \(y\) in \(G\) satisfying \(x*y = e\) and \(y*x=e\).

Definition \(\PageIndex{3}\): Abelian

A group \((G,*)\) is said to be Abelian if \(x*y=y*x\) for all \(x,y \in G\).

Thus, to describe a group one must specify two things:

1. a set, and
2. a binary operation on the set.

Then, one must verify that the binary operation is associative, that there is an identity in the set, and that every element in the set has an inverse.

Example \(\PageIndex{2}\)

Here are some examples of groups. The binary operations are in each case the ones that you are familiar with.

1. \((\mathbb{Z},+)\) is a group with identity 0. The inverse of \(x \in \mathbb{Z}\) is \(-x\).
2. \((\mathbb{Q},+)\) is a group with identity 0. The inverse of \(x \in \mathbb{Q}\) is \(-x\).
3. \((\mathbb{R},+)\) is a group with identity 0. The inverse of \(x \in \mathbb{R}\) is \(-x\).
4. \((\mathbb{Q}-\{0\},\cdot)\) is a group with identity 1. The inverse of \(x \in \mathbb{Q}-\{0\}\) is \(x^{-1}\).
5. \((\mathbb{R}-\{0\},\cdot)\) is a group with identity 1. The inverse of \(x \in \mathbb{R}-\{0\}\) is \(x^{-1}\).
6. \((\mathbb{Z}_n,+)\) is a group with identity 0. The inverse of \(x \in \mathbb{Z}_n\) is \(n-x\) if \(x \ne 0\), the inverse of 0 is 0.
7. \((U_n,\cdot )\) is a group with identity \([1]\). The inverse of \([a] \in U_n\) was shown to exist in Chapter 22.
8. \((\mathbb{R}^n,+)\) where \(+\) is vector addition. The identity is the zero vector \((0,0,\dots,0)\) and the inverse of the vector \(\mathbf{x}=(x_1,x_2,\dots,x_n)\) is the vector \(\mathbf{-x}=(-x_1,-x_2,\dots,-x_n)\).
9. \((M_n(\mathbb{R}),+)\). This is the group of all \(n \times n\) matrices over \(\mathbb{R}\) and \(+\) is matrix addition.

Bibliography

[1] Tom Apostol, Introduction to Analytic Number Theory, Springer-Verlag, New York-Heidelberg, 1976.

[2] Chris Caldwell, The Primes Pages,
http://www.utm.edu/research/primes/

[3] W. Edwin Clark, Number Theory Links,
http://www.math.usf.edu/~eclark/numtheory_links.html

[4] Earl Fife and Larry Husch, Number Theory (Mathematics Archives,
http://archives.math.utk.edu/topics/numberTheory.html

[5] Ronald Graham, Donald Knuth, and Oren Patashnik, Concrete Mathematics, Addison-Wesley, 1994.

[6] Donald Knuth The Art of Computer Programming, Vols I and II, Addison-Wesley, 1997.

[7] The Math Forum, Number Theory Sites
http://mathforum.org/library/topics/number_theory/

[8] Oystein Ore, Number Theory and its History, Dover Publications, 1988.

[9] Carl Pomerance and Richard Crandall, Prime Numbers – A Computational Perspective, Springer -Verlag, 2001.

[10] Kenneth A. Rosen, Elementary Number Theory, (Fourth Edition), Addison-Wesley, 2000.

[11] Eric Weisstein, World of Mathematics –Number Theory Section,
http://mathworld.wolfram.com/topics/NumberTheory.html