﻿ ﻿RINGS: DEFINITIONS AND ELEMENTARY PROPERTIES - A Book of Abstract Algebra

## A Book of Abstract Algebra, Second Edition (1982)

### Chapter 17. RINGS: DEFINITIONS AND ELEMENTARY PROPERTIES

In presenting scientific knowledge it is elegant as well as enlightening to begin with the simple and move toward the more complex. If we build upon a knowledge of the simplest things, it is easier to understand the more complex ones. In the first part of this book we dedicated ourselves to the study of groups—surely one of the simplest and most fundamental of all algebraic systems. We will now move on, and, using the knowledge and insights gained in the study of groups, we will begin to examine algebraic systems which have two operations instead of just one.

The most basic of the two-operational systems is called a ring: it will be defined in a moment. The surprising fact about rings is that, despite their having two operations and being more complex than groups, their fundamental properties follow exactly the pattern already laid out for groups. With remarkable, almost compelling ease, we will find two-operational analogs of the notions of subgroup and quotient group, homomorphism and isomorphism—as well as other algebraic notions— and we will discover that rings behave just like groups with respect to these notions.

The two operations of a ring are traditionally called addition and multiplication, and are denoted as usual by + and ·, respectively. We must remember, however, that the elements of a ring are not necessarily numbers (for example, there are rings of functions, rings of switching circuits, and so on); and therefore “addition” does not necessarily refer to the conventional addition of numbers, nor does multiplication necessarily refer to the conventional operation of multiplying numbers. In fact, + and · are nothing more than symbols denoting the two operations of a ring.

By a ring we mean a set A with operations called addition and multiplication which satisfy the following axioms:

(i)A with addition alone is an abelian group.

(ii)Multiplication is associative.

(iii)Multiplication is distributive over addition. That is, for all a, b, and c in A,

a(b + c) = ab + ac

and

(b + c)a = ba + ca

Since A with addition alone is an abelian group, there is in A a neutral element for addition: it is called the zero element and is written 0. Also, every element has an additive inverse called its negative; the negative of a is denoted by −a. Subtraction is defined by

ab = a + (−b)

The easiest examples of rings are the traditional number systems. The set of the integers, with conventional addition and multiplication, is a ring called the ring of the integers. We designate this ring simply with the letter . (The context will make it clear whether we are referring to the ring of the integers or the additive group of the integers.)

Similarly, is the ring of the rational numbers, the ring of the real numbers, and the ring of the complex numbers. In each case, the operations are conventional addition and multiplication.

Remember that () represents the set of all the functions from to ; that is, the set of all real-valued functions of a real variable. In calculus we learned to add and multiply functions: if f and g are any two functions from to , their sum f + g and their product fg are defined as follows:

[f + g](x) = f(x) + g(x) for every real number x

and

[fg](x) = f(x)g(x) for every real number x

() with these operations for adding and multiplying functions is a ring called the ring of real functions. It is written simply as (). On page 46 we saw that () with only addition of functions is an abelian group. It is left as an exercise for you to verify that multiplication of functions is associative and distributive over addition of functions.

The rings , , ,, and () are all infinite rings, that is, rings with infinitely many elements. There are also finite rings: rings with a finite number of elements. As an important example, consider the group n, and define an operation of multiplication on n by allowing the product ab to be the remainder of the usual product of integers a and b after division by n. (For example, in 5, 2 · 4 = 3, 3 · 3 = 4, and 4·3 = 2.) This operation is called multiplication modulo n. n with addition and multiplication modulo η is a ring: the details are given in Chapter 19.

Let A be any ring. Since A with addition alone is an abelian group, everything we know about abelian groups applies to it. However, it is important to remember that A with addition is an abelian group in additive notation and, therefore, before applying theorems about groups to A, these theorems must be translated into additive notation. For example, Theorems 1, 2, and 3 of Chapter 4 read as follows when the notation is additive and the group is abelian:

a + b = a + c implies b = c (1)

a + b = 0 impliesa = −b and b = −a (2)

−(a + b) = −(a) + −(b) and − (−a) = a (3)

Therefore Conditions (1), (2), and (3) are true in every ring.

What happens in a ring when we multiply elements by zero? What happens when we multiply elements by the negatives of other elements? The next theorem answers these questions.

Theorem 1 Let a and b be any elements of a ring A.

(i)a0 = 0 and 0a = 0

(ii)a(−b) = − (ab) and (−a)b = −(ab)

(iii)(−a)(−b) = ab

Part (i) asserts that multiplication by zero always yields zero, and parts (ii) and (iii) state the familiar rules of signs.

PROOF: To prove (i) we note that

Thus, aa + 0 = aa + a0. By Condition (1) above we may eliminate the term aa on both sides of this equation, and therefore 0 = a0.

To prove (ii), we have

Thus, a(−b) + ab = 0. By Condition (2) above we deduce that a(−b) = − (ab). The twin formula (−a)b = − (−ab) is deduced analogously.

We prove part (iii) by using part (ii) twice:

(−a)(−b) = −[a(−b)] = −[ − (ab)] = ab

The general definition of a ring is sparse and simple. However, particular rings may also have “optional features” which make them more versatile and interesting. Some of these options are described next.

By definition, addition is commutative in every ring but mutiplication is not. When multiplication also is commutative in a ring, we call that ring a commutative ring.

A ring A does not necessarily have a neutral element for multiplication. If there is in A a neutral element for multiplication, it is called the unity of A, and is denoted by the symbol 1. Thus, a . 1 = a and 1 . a = a for every a in A. If A has a unity, we call A a ring with unity. The rings ,, , , and () are all examples of commutative rings with unity.

Incidentally, a ring whose only element is 0 is called a trivial ring; a ring with more than one element is nontrivial. In a nontrivial ring with unity, necessarily 1 ≠ 0. This is true because if 1 = 0 and x is any element of the ring, then

x = x1 = x0 = 0

In other words, if 1 = 0 then every element of the ring is equal to 0; hence 0 is the only element of the ring.

If A is a ring with unity, there may be elements in A which have a multiplicative inverse. Such elements are said to be invertible. Thus, an element a is invertible in a ring if there is some x in the ring such that

ax = xa = 1

For example, in every nonzero element is invertible: its multiplicative inverse is its reciprocal. On the other hand, in the only invertible elements are 1 and −1.

Zero is never an invertible element of a ring except if the ring is trivial; for if zero had a multiplicative inverse x, we would have 0x = 1, that is, 0 = 1.

If A is a commutative ring with unity in which every nonzero element is invertible, A is called a field. Fields are of the utmost importance in mathematics; for example, ,, and are fields. There are also finite fields, such as 5(it is easy to check that every nonzero element of 5 is invertible). Finite fields have beautiful properties and fascinating applications, which will be examined later in this book.

In elementary mathematics we learned the commandment that if the product of two numbers is equal to zero, say

ab = 0

then one of the two factors, either a or b (or both) must be equal to zero. This is certainly true if the numbers are real (or even complex) numbers, but the rule is not inviolable in every ring. For example, in 6,

2·3 = 0

even though the factors 2 and 3 are both nonzero. Such numbers, when they exist, are called divisors of zero.

In any ring, a nonzero element a is called a divisor of zero if there is a nonzero element b in the ring such that the product ab or ba is equal to zero.

(Note carefully that both factors have to be nonzero.) Thus, 2 and 3 are divisors of zero in 6; 4 is also a divisor of zero in 6, because 4·3 = 0. For another example, let 2() designate the set of all 2 × 2 matrices of real numbers, with addition and multiplication of matrices as described on page 8. The simple task of checking that 2() satisfies the ring axioms is assigned as Exercise C1 at the end of this chapter. 2() is rampant with examples of divisors of zero. For instance,

hence

are both divisors of zero in 2().

Of course, there are rings which have no divisors of zero at all! For example, ,, , and do not have any divisors of zero. It is important to note carefully what it means for a ring to have no divisors of zero: it means that if the product of two elements in the ring is equal to zero, at least one of the factors is zero. (Our commandment from elementary mathematics!)

It is also decreed in elementary algebra that a nonzero number a may be canceled in the equation ax = ay to yield x = y. While undeniably true in the number systems of mathematics, this rule is not true in every ring. For example, in 6,

2.5 = 2.2

yet we cannot cancel the common factor 2. A similar example involving 2×2 matrices may be seen on page 9. When cancellation is possible, we say the ring has the “cancellation property.”

A ring is said to have the cancellation property if

ab = ac or ba = ca implies b = c

for any elements a, b, and c in the ring if a ≠ 0.

There is a surprising and unexpected connection between the cancellation property and divisors of zero:

Theorem 2 A ring has the cancellation property iff it has no divisors of zero.

PROOF: The proof is very straightforward. Let A be a ring, and suppose first that A has the cancellation property. To prove that A has no divisors of zero we begin by letting ab = 0, and show that a or b is equal to 0. If a = 0, we are done. Otherwise, we have

ab = 0 = a0

so by the cancellation property (cancelling a), b=0.

Conversely, assume A has no divisors of zero. To prove that A has the cancellation property, suppose ab = ac where a ≠ 0. Then

abac = a(bc) = 0

Remember, there are no divisors of zero! Since a ≠ 0, necessarily bc = 0, so b = c.■

An integral domain is defined to be a commutative ring with unity having the cancellation property. By Theorem 2, an integral domain may also be defined as a commutative ring with unity having no divisors of zero. It is easy to see that every field is an integral domain. The converse, however, is not true: for example, is an integral domain but not a field. We will have a lot to say about integral domains in the following chapters.

EXERCISES

A. Examples of Rings

In each of the following, a set A with operations of addition and multiplication is given. Prove that A satisfies all the axioms to be a commutative ring with unity. Indicate the zero element, the unity, and the negative of an arbitrary a.

1 A is the set of the integers, with the following “addition” ⊕ and “multiplication” :

ab = a + b−1 a b = ab − (a + b) + 2

2 A is the set of the rational numbers, and the operations are ⊕ and defined as follows:

ab = a + b + 1 a b = ab + a + b

# 3 A is the set × of ordered pairs of rational numbers, and the operations are the following addition ⊕ and multiplication :

4 with conventional addition and multiplication.

5 Prove that the ring in part 1 is an integral domain.

6 Prove that the ring in part 2 is a field, and indicate the multiplicative inverse of an arbitrary nonzero element.

7 Do the same for the ring in part 3.

B. Ring of Real Functions

1 Verify that () satisfies all the axioms for being a commutative ring with unity. Indicate the zero and unity, and describe the negative of any f.

# 2 Describe the divisors of zero in ().

3 Describe the invertible elements in ().

4 Explain why () is neither a field nor an integral domain.

C. Ring of 2 × 2 Matrices

Let 2() designate the set of all 2 × 2 matrices

whose entries are real numbers a, b, c, and d, with the following addition and multiplication:

and

1Verify that 2() satisfies the ring axioms.

2Show that 2() is not commutative and has a unity.

3Explain why 2() is not an integral domain or a field.

D. Rings of Subsets of a Set

If D is a set, then the power set of D is the set PD of all the subsets of D. Addition and multiplication are defined as follows: If A and B are elements of PD (that is, subsets of D), then

A + B = (AB)∪(BA) and AB = AB

It was shown in Chapter 3, Exercise C, that PD with addition alone is an abelian group.

# 1 Prove: PD is a commutative ring with unity. (You may assume ∩ is associative; for the distributive law, use the same diagram and approach as was used to prove that addition is associative in Chapter 3, Exercise C.)

2 Describe the divisors of zero in PD.

3 Describe the invertible elements in PD.

4 Explain why PD is neither a field nor an integral domain. (Assume D has more than one element.)

5 Give the tables of P3, that is, PD where D = {a, b, c} .

E. Ring of Quaternions

A quaternion(in matrix form) is a 2 × 2 matrix of complex numbers of the form

1 Prove that the set of all the quaternions, with the matrix addition and multiplication explained on pages 7 and 8, is a ring with unity. This ring is denoted by the symbol . Find an example to show that is not commutative. (You may assume matrix addition and multiplication are associative and obey the distributive law.)

2 Let

Show that the quaternion a, defined previously, may be written in the form

α = al + bi + cj + dk

(This is the standard notation for quaternions.)

# 3 Prove the following formulas:

i2 = j2 = k2 = −l ij = −ji = k jk=−kj = i ki=−ik=j

4 The conjugate of α is

The norm of α is a2 + b2 + c2 + d2, and is written ∥α∥.Show directly (by matrix multiplication) that

Conclude that the multiplicative inverse of α is (1/t) .

5 A skew field is a (not necessarily commutative) ring with unity in which every nonzero element has a multiplicative inverse. Conclude from parts 1 and 4 that is a skew field.

F. Ring of Endomorphisms

Let G be an abelian group in additive notation. An endomorphism of G is a homomorphism from G to G. Let End(G) denote the set of all the endomorphisms of G, and define addition and multiplication of endomorphisms as follows:

1 Prove that End(G) with these operations is a ring with unity.

2 List the elements of End(4), then give the addition and multiplication tables for End(4).

REMARK: The endomorphisms of 4 are easy to find. Any endomorphisms of 4 will carry 1 to either 0, 1, 2, or 3. For example, take the last case: if

then necessarily

hence f is completely determined by the fact that

G. Direct Product of Rings

If A and B are rings, their direct product is a new ring, denoted by A × B, and defined as follows: A × B consists of all the ordered pairs (x, y) where x is in A and y is in B. Addition in A × B consists of adding corresponding components:

(x1, y1) + (x2, y2) = (x1+x2, y1+y2)

Multiplication in A × B consists of multiplying corresponding components:

(x1, y1) · (x2, y2) = (x1x2, y1y2)

1 If A and B are rings, verify that A × B is a ring.

2 If A and B are commutative, show that A × Bis commutative. If A and B each has a unity, show that A × B has a unity.

3 Describe carefully the divisors of zero in A × B.

# 4 Describe the invertible elements in A × B.

5 Explain why A × B can never be an integral domain or a field. (Assume A and B each have more than one element.)

H. Elementary Properties of Rings

Prove parts 1−4:

1 In any ring, a(bc) = abac and (bc)a = baca.

2 In any ring, if ab = −ba, then (a + b)2 = (ab)2 = a2 + b2.

3 In any integral domain, if a2 = b2, then a = ±b.

4 In any integral domain, only 1 and −1 are their own multiplicative inverses. (Note that x = x1 iff x2 = 1.)

5 Show that the commutative law for addition need not be assumed in defining a ring with unity: it may be proved from the other axioms. [HINT: Use the distributive law to expand (a + b)(1 + 1) in two different ways.]

# 6 Let A be any ring. Prove that if the additive group of A is cyclic, then A is a commutative ring.

7 Prove: In any integral domain, if an = 0 for some integer n, then a = 0.

I. Properties of Invertible Elements

Prove that parts 1−5 are true in a nontrivial ring with unity.

1 If a is invertible and ab = ac, then b = c.

2 An element a can have no more than one multiplicative inverse.

3 If a2 = 0 then a + 1 and a − 1 are invertible.

4 If a and b are invertible, their product ab is invertible.

5 The set S of all the invertible elements in a ring is a multiplicative group.

6 By part 5, the set of all the nonzero elements in a field is a multiplicative group. Now use Lagrange’s theorem to prove that in a finite field with m elements, xm1 = 1 for every x ≠ 0.

7 If ax = 1, x is a right inverse of a; if ya = 1, y is a left inverse of a. Prove that if a has a right inverse y and a left inverse y, then a is invertible, and its inverse is equal to x and to y. (First show that yaxa = 1.)

8 Prove: In a commutative ring, if ab is invertible, then a and b are both invertible.

J. Properties of Divisors of Zero

Prove that each of the following is true in a nontrivial ring.

1 If a ≠ ±1 and a2 = 1, then a + 1 and a − 1 are divisors of zero.

# 2 If ab is a divisor of zero, then a or b is a divisor of zero.

3 In a commutative ring with unity, a divisor of zero cannot be invertible.

4 Suppose ab ≠ 0 in a commutative ring. If either α or is a divisor of zero, so is ab.

5 Suppose a is neither 0 nor a divisor of zero. If ab = ac, then b = c.

6 A × B always has divisors of zero.

K. Boolean Rings

A ring A is a boolean ring if a2 = a for every aA. Prove that parts 1 and 2 are true in any boolean ring A.

1For every aA, a = −a. [HINT: Expand (a + a)2.]

2Use part 1 to prove that A is a commutative ring. [HINT: Expand (a + b)2.]

In parts 3 and 4, assume A has a unity and prove:

3Every element except 0 and 1 is a divisor of zero. [Consider x(x − 1).]

41 is the only invertible element in A.

5Letting ab = a + b + ab we have the following in A:

abc = (ab)(ac) a∨ (1 + a) = 1 aa = a a(ab) = a

L. The Binomial Formula

An important formula in elementary algebra is the binomial expansion formula for an expression (a + b)n. The formula is as follows:

where the binomial coefficient

This theorem is true in every commutative ring. (If K is any positive integer and a is an element of a ring, ka refers to the sum a + a + ⋯ + a with k terms, as in elementary algebra.) The proof of the binomial theorem in a commutative ring is no different from the proof in elementary algebra. We shall review it here.

The proof of the binomial formula is by induction on the exponent n. The formula is trivially true for n = 1. In the induction step, we assume the expansion for (a + b)n is as above, and we must prove that

Now,

Collecting terms, we find that the coefficient of an + 1 kbk is

By direct computation, show that

It will follow that (a + b)n + 1 is as claimed, and the proof is complete.

M. Nilpotent and Unipotent Elements

An element a of a ring is nilpotent if an = 0 for some positive integer n.

1 In a ring with unity, prove that if a is nilpotent, then a +1 and a − 1 are both invertible. [HINT: Use the factorization

1 − an = (1 − a)(1 + a + a2 + ⋯ + a n1)

for 1 − a, and a similar formula for 1 + a.]

2 In a commutative ring, prove that any product xa of a nilpotent element a by any element x is nilpotent.

# 3 In a commutative ring, prove that the sum of two nilpotent elements is nilpotent. (HINT: You must use the binomial formula; see Exercise L.)

An element a of a ring is unipotent iff 1 − a is nilpotent.

4 In a commutative ring, prove that the product of two unipotent elements a and b is unipotent. [HINT: Use the binomial formula to expand 1 − ab = (1 − a) + a(1 − b) to power n + m.]

5 In a ring with unity, prove that every unipotent element is invertible. (HINT: Use Part 1.)

﻿