For example, if f(x) = x, g(x) = 2x², we have (f + g)(x) = x + 2x², and (f · g)(x) = 2x³. Note that we have defined the sum and product of functions by specifying the values of the sum and products for all values x R. Thus, by the sum and product functions in R also belong to R, and we have so defined two binary operations on R. With respect to these operations, R is a ring. We have already verified earlier that with respect to +, R is an abelian group. Thus it suffices to check axioms R2 and R3, which is fairly simply and we will leave to the reader. It is easy to see that R is a commutative ring with identity. The identity element is the function I defined by I(x) = 1 for all x R, Note that in example 6, it can happen that the product of two nonzero elements of R is the zero element of R. For the zero element of R is the function 0 defined by 0(x) = 0 for all x R. Let f, g R be defined by

Then f 0, g 0, but f · g = 0.

Example 8: Let R be the set of all functions f :RR of the form

Such a function is called a polynomial function. Let addition and multiplication of functions be defined as in Example 7. Then the sum (and product) of two polynomial functions is again a polynomial function, so we have two binary operations defined on R. With respect to these operations, R is a commutative ring with identity.

Example 9: Let R denote the set of all 2 2 matrices with real entries. A typical element of R looks like

where a, b,c, d, R. Let us define addition and multiplication of matrices by

With respect to these operations, R becomes a ring with identity. The zero element and the identity of R are given by

Note that

Therefore, R is not a commutative ring. We will henceforth denote the ring of 2 2 matrices with real entries by M₂(R). Note that the last computation also shows that the product of two nonzero matrices may be zero.

Example 10: Let d be an integer, and let R consist of the set of all complex numbers of the form a + b, where a, b Z. Let addition and multiplication in R be the usual addition and multiplication for complex numbers, respectively. Since

we see that addition and multiplication define binary operations on R. The fact that R forms a ring is now trivial to verify from the properties of the complex numbers. The ring R will be denoted Z[].

Example 11: Let R be the set of all complex numbers of the form a + b, where d is given and a, b Q. Then the ring R is a ring which contains Z[]. We will denote the ring of this example by Q().

Example 12: Let R and S be any two rings, and let R S denote the Cartesian product of R and S as sets). Let us define addition and multiplication in R S by

Then it is easy to verify that R S is a ring, called the Cartesian product of R and S. Sometimes, the Cartesian product is called the direct sum of R and S and is denoted by R S.

Now that we have surveyed the wide possibilities, which the ring axioms create, let us establish some elementary properties of rings. The following proposition will start us along that line.

Proposition 4: Let R be a ring. a, b R. Then

(1) a · 0 = 0 · a = 0.

(2) a · (-b) = (-a) · b = -(a · b).

(3) (-a) · (-b) = a · b.

Note that these rules of arithmetic in an arbitrary ring are just generalizations of well-known facts from the algebra of the integers, Z.

Proof:

(1) Since 0 is the identity with respect to addition, 0 + 0 = 0. Therefore, by the distributive law,

Therefore, by adding -(a · 0) to both sides of this last equation, we get a · 0 = 0. Similarly 0 · a = 0.

(2) Let y = -(a · b). Then a · b + y = 0. Therefore,

(-a) · b = (-a) · b + 0

= (-a) · b + [a · b + y]

= (-a + a) · b + y   (by associative and distributive laws)

= 0 · b + y   (since -a + a = 0)

= 0 + y   (by (1))

= y.

Therefore, (-a) · b = -(a · b).

(3) By part (2),

(-a) · (-b) = -a · (-b)

= -(-a · b)

= a · b   (since R is a group with respect to addition)

Let us now examine some of the phenomena which were exhibited by our examples. In Examples 6 and 7 we found that there exist nonzero ring elements a and b such that a · b = 0. Note that if a or b is zero, then a · b = 0 by Part (1) above. However, it is somewhat surprising that nonzero elements can have a zero product. Therefore, we are prompted to make the following definition.

Definition 5: Let R be a ring, a R, a 0. We say that a is a right zero divisor (respectively, left zero divisor) if there exists b R, b 0, such that a · b = 0 (respectively, b · a = 0). An element of R is called a zero divisor of R if it is either a right or left zero divisor.

If R is a commutative ring, then the distinction between right and left zero divisors disappears. An element of a commutative ring R is a right zero divisor if and only if it is a left zero divisor.

Definition 6: A commutative ring R is said to be an integral domain if R contains no zero divisors. In other words, R is an integral domain if the product of any two nonzero elements of R is nonzero. (For convenience in stating various results, we will always assume that an integral domain always contains an identity.)

For example Z₆ is not an integral domain because 2 is a zero divisor. Similarly, Example 7 shows the ring of functions f :RR is not an integral domain. On the other hand Z is an integral domain. For if a and b are nonzero integers, then a · b 0. Moreover if p is prime, then Z_p is an integral domain.

Proposition 7: Suppose that R is an integral domain, a,b,c R, a 0. Suppose that a · b = a · c. Then b = c.

Proof: By Proposition 4 and the distributive law, we have

However, since a 0 and R is an integral domain, b - c = 0, and thus b = c.

If R is a ring with identity 1, it makes sense to ask whether an element a of R has an inverse with respect to multiplication. That is, does there exist b R such that

If such an element b exists, it is unique. We have already worked through this sort of argument several times, and therefore we will leave the verification that b is unique as an exercise for the reader. If a has an inverse with respect to multiplication, then the unique inverse will be denoted a^-1.

Definition 8: Let R be a ring with identity 1, and let a R. If a has an inverse with respect to multiplication, then we say that a is a unit of R. (Some texts will also say that a is invertible.)

Example 13: Let R be any ring with identity. Then 1 and -1 are units. In fact, since 1 · 1 = 1, (-1) · (-1) = 1 · 1 = 1, we see that

Example 14: Let R = Z. Then the only units of R are 1 and -1. For if a R is a unit, there exists b R such that a · b = 1. Then a = b = 1 or a = b = -1

Example 15: Let R = Q. Then 0 is not a unit of R since 0 · b = 0 1 for all b R. However, if a/b Q - {0}, then a/b is a unit of R. In fact, since

we see that (a/b)^-1 = b/a.

Example 16: Let R = the trivial ring. Then, since 1 = 0 in this ring, 0 is a unit.

Example 17: Let R be any ring with identity which is not the trivial ring. In this ring 1 0. For if R is not the trivial ring, there exists a nonzero element in R. Pick one such element a. Then, if 1 = 0, we would have

However, 1 · a = a 0 Thus we reach a contradiction, so that 1 0. Let us now show that 0 is not a unit of R. Indeed, for all b R, we have 0 · b = 0 1. Thus in any ring that is not the trivial ring, 0 is not a unit.

Example 18: Let R = {a + b | a,b Z} be the ring of Example 10. The units of R are not quite so easy to write down as the following special cases show:

d = -1: 1 + 0, -1 + 0, 0 + , 0 - are units with respective inverses 1 + 0, -1 + 0, 0 - , 0 + .

d = 2: 1 + 0, -1 + 0, 1 + , 1 - , 3 + 2, 3 - 2, -3 + 2, -3 - 2 are unit with respect to inverses 1 + 0, -1 + 0, -1 + , -1 - , 3 - 2, 3 + 2, -3 - 2, -3 + 2 [All these units are powers of ±(1 + )ⁿ for various values of n.]

It is worthwhile to record the observation made in example 17 as a proposition.

Proposition 19: Let R be a ring with identity which is not the trivial ring. Then 1_R 0_R.

For any ring with identity, let us denote by U_R the set of all units of R. From our above examples, we know that 1, -1 U_R and that if R is not the trivial ring, then 0 U_R.

Proposition 20: Let R be a ring with identity, and let a, b U_R.

(1) a^-1 U_R.

(2) a · b U_R and (a · b)^-1 = b^-1 · a^-1.

(3) U_R is a group with respect to multiplication.

Proof:

(1) Since a U_R, a has an inverse a^-1 and a·a^-1 = a^-1·a = 1. But these last equations imply that a^-1 is a unit of R and that its inverse is a. Thus, we have proven part (1).

(2) Since a · a^-1 = a^-1 · a = 1, b · b^-1 = b^-1 · b = 1, we see that

These last two equations show how a · b is a unit of R and that its inverse is b^-1·a^-1. This proves part (2)

(3) By part (2), U_R is closed with respect to multiplication. Moreover, we observed above that 1 U_R, so that U_R has an identity with respect to multiplication. Multiplication in U_R is associative since multiplication in R is associative. Finally, by part (1), if a U_R, then a has an inverse with respect to multiplication in U_R, namely a_-1. Thus, U_R satisfies the group axioms.

The group U_R is called the group of units of R.

In order to introduce the last of special classes of rings which we will examine in this section, let us confine our discussion to a ring R with identity which is not the trivial ring. Let us denote the subset R - {0} of R by R^x. We showed earlier that 0 is not a unit of R. Therefore U_R R^x. Note that it is not always true that U_R = R^x. For example, we saw that if R = Z, then U_R = {1, -1}. However, we also found that if R = Q, then U_R = Q - {0} = R^x. Thus for some rings we have U_R = R^x. Such rings play an exceeding important role in algebra, for reasons we will discuss below. Before we give this class of rings a name, let us observe that U_R = R^x if and only if every nonzero element of R has an inverse with respect to multiplication. Thus let us make the following definition.

Definition 21: A field is a nontrivial commutative ring with identity in which every nonzero element has an inverse with respect to multiplication.

Note that a field must contain at least two elements, since it is a nontrivial ring. Therefore, in particular, in a field, we have 1 0. Note also that we have assumed that a field is commutative. Some authors define a field without this condition. And, indeed, there are many systems which satisfy the axioms for a field, with the exception of commutativity. Such systems are called skew fields, division algebras, or division rings, all of these terms being used interchangeably. In general we will not be discussing any of these object for the remainder of this document. We will assume the commutativity of a field unless otherwise stated.

Example 22: A a result of our previous discussion, we see that Q is a field.

Example 23: R, the set of real numbers, is a field.

Example 24: Let p be a prime and let Z_p denote the ring of residue classes modulo p. Recall that Z_p = {0,1, ..., p-1}. Since p is a prime, we have (i,p) = 1 for i = 1,2,...,p-1. Therefore, each of the residue classes 1,2,...,p-1 is reduced. Therefore, since 0 is clearly not reduced, we have Z_p^x = {1, 2, ..., p-1}. That is every nonzero element of Z_p is a reduced residue class. We showed in the previous section on congruences, that every reduced residue class has an inverse with respect to multiplication in Z_p. Therefore, Z_p is a field. In fact Z_p is a field with p elements. A field with a finite number of elements is called a finite field. Therefore, we can summarize this example by saying that Z_p is a finite field.

Proposition 25: Let F be a nontrivial commutative ring with identity. Then F is a field if and only if F^x is a group with respect to multiplication.

Proof: . Assume that F is a field. Then by definition of a field F^x = U_F. However, we have shown that U_F is a group. Therefore, F^x is a group.

. Assume that F^x is a group. Then every element of F^x has an inverse with respect to multiplication. Therefore, F is a field.

Corollary 26: Let F be a field. Then F is an integral domain.

Proof: If F is a field, then F^X is a group. F^x is closed with respect to multiplication. Therefore, F is an integral domain.

The extreme importance of fields in mathematics is due to the fact that in a field it is possible to combine elements using all the operations which one meets in elementary arithmetic. That is, given any two elements a and b of a field, it is possible to form their sum (a + b), difference (a - b), product (a · b), and, if b 0, their quotient (a · b^-1.

If R is a ring with identity, then it is convenient to introduce the language of exponents in R. Let a R, n is a nonnegative integer. Then we define aⁿ inductively by

Then the following laws of exponents hold:

(1) (2)

Moreover, if R is commutative, then

(3)

Another useful definition is:

Definition 27:The characteristic of a ring R (written Char R) is the smallest positive integer such that n1 = 0,where n1 is an abbreviation for 1 + 1 + ... + 1 (n times). If n1 is never 0,we say that R has characteristic 0. Note that the characteristic can never be 1,since 1_R 0. If R is an integral domain and Char R 0, then Char R must be a prime number. For if Char R = n = rs where r and s are positive integers greater than 1,then (r1)(s1) = n1 = 0, so either r1 or s1 is 0,contradicting the minimality of n.