3.1 Rings and fields

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A ring $R$ is a set with two laws of composition called addition and multiplication, denoted $a+b$ and $ab$ respectively. It is required that $R$ is an abelian group with respect to $+$, with identity element $0$ and inverses denoted $-a$. With respect to multiplication $R$ is to be a commutative semigroup, so that the identity and inverses are not necessarily present. In detail, the requirements of a ring are:

(R1) Addition is associative, $(a+b)+c=a+(b+c)$

(R2) Addition is commutative, $a+b=b+a.$

(R3) There is an element 0 such that $a+0=a$ for all $a\in R$

(R4) For each $a\in R$ there exists an element $-a$ such that $a-a\equiv a+(-a)=0.$

(R5) Multiplication is associative, $(ab)c=a(bc)$

(R6) Multiplication is commutative, $ab=ba.$

(R7) The distributive law holds, $a(b+c)=ab+ac$ . By (R6) this also implies $(a+b)c=ac+bc$ . It is the key relation linking the two laws of composition, addition and multiplication.

As shown in Chapter 2, the additive identity 0 is unique. From these axioms we also have that $0a=0$ for all $a\in R.$ , for by (R1), $R$(3), (R4) and (R7)

$$0a=0a+0=0a+0a-0a=(0+0)a-0a=0a-0a=0.$$

Example 3.1

The integers $\mathbb{Z}$ form a ring with respect to the usual operation ofaddition and multiplication. This ring has a (multiplicative) identity 1, having the property $1a=a1=a$ for all $a\in \mathbb{Z}$ . The set $2\mathbb{Z}$ consisting of all even integers also forms a ring, but now there is no identity.

Example 3.2

The set $M_n$ of all $n\times n$ real matrices forms a ring with addition of matrices $\mathsf{A}+\mathsf{B}$ and matrix product $AB$ defined in the usual way. This is a ring with identity $I$, the unit matrix.

Example 3.3

The set of all real-valued functions on a set $S$, denoted $\mathcal{F}(S)$ , forms a ring with identity. Addition and multiplication of functions $f+g$ $fg$ are defined in the usual way,

$$(f+g)(x)=f(x)+g(x),(fg)(x)=f(x)g(x).$$

The 0 element is the zero function whose value on every $x\in S$ is the number zero, while the identity is the function having the value 1 at each $x\in S$

These examples of rings all fail to be groups with respect to multiplication, for even when they have a multiplicative identity 1, it is almost never true that the zero element 0 has an inverse.

Exercise

Show that if $0^{-1}$ exists in a ring $R$ with identity then $0=1$ and $R$ must be the trivial ring consisting of just one element 0.

A field $\mathbb{K}$ is a ring with a multiplicative identity 1, in which every element $a\ne 0$ has an inverse $a^{-1}\in \mathbb{K}$ such that $a{a}^{-1}=1$ . It not totally clear why the words ‘rings’ and ‘fields’ are used to describe these algebraic entities. However, the word ‘field’ is a perhaps a little unfortunate as it has nothing whatsoever to do with expressions such as ‘electromagnetic field’, commonly used in physics.

Example 3.4

The real numbers $R$ and complex numbers $C$ both form fields with respect to the usual rules of addition and multiplication. These are essentially the only fields of interest in this book. We will frequently use the symbol $\mathbb{K}$ to refer to a field which could be either $R$ or $C$.

Problems

Problem 3.1

Show that the integers modulo a prime number $p$ form a finite field.

Problem 3.2

Show that the set of all real numbers of the form $a+b\sqrt{2},$ , where $a$ and $b$ are rational numbers, is a field. If $a$ and $b$ are restricted to the integers show that this set is a ring, but is not a field.