2.3: 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 the webpage:

 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 \(\PageIndex{1}\)

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 \(\PageIndex{3}\)

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 Section 1.22.
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.