Schaum's Outline of Discrete Mathematics, Third Edition (Schaum's Outlines)

444 ALGEBRAIC SYSTEMS [APP. B

Special Kinds of Rings: Integral Domains and Fields

This subsection defines a number of different kinds of rings, including integral domains and fields.
Ris called acommutative ringifab=bafor everya,b∈R.
Ris called aring with an identity element 1if the element 1 has the property thata· 1 = 1 ·a=afor every
elementa∈R. In such a case, an elementa∈Ris called aunitifahas a multiplicative inverse, that is, an
elementa−^1 inRsuch thata·a−^1 =a−^1 ·a=1.
Ris called aring with zero divisorsif there exist nonzero elementsa, b∈Rsuch thatab=0. In sucha
case,aandbare calledzero divisors.

Definition B.3: A commutative ringRis anintegral domainifRhas no zero divisors, that is, ifab=0 implies
a=0orb=0.

Definition B.4: A commutative ringRwith an identity element 1 (not equal to 0) is a field if every nonzero
a∈Ris a unit, that is, has a multiplicative inverse.

A field is necessarily an integral domain; for ifab=0 anda=0, then

b= 1 ·b=a−^1 ab=a−^1 · 0 = 0

We remark that a field may also be viewed as a commutative ring in which the nonzero elements form a group
under multiplication.

EXAMPLE B.13

(a) The setZof integers with the usual operations of addition and multiplication is the classical example of an

integral domain (with an identity element). The units inZare only 1 and−1, that is, no other element inZ

has a multiplicative inverse.

(b) The setZm={ 0 , 1 , 2 ,...,m− 1 }under the operation of addition and multiplication modulomis a ring;

it is called thering of integers modulo m.Ifmis a prime, thenZmis a field. On the other hand, ifmis

not a prime thenZmhas zero divisors. For instance, in the ringZ 6 ,

2 · 3 =0 but 2≡ 0 (mod 6) and 3≡ 0 (mod 6)

(c) The rational numbersQand the real numbersReach form a field with respect to the usual operations of

addition and multiplication.

(d) LetMdenote the set of 2×2 matrices with integer or real entries. ThenMis a noncommutative ring with zero

divisors under the operations of matrix addition and matrix multiplication.Mdoes have an identity element,

the identity matrix.

(e) LetRbe any ring. Then the setR[x]of all polynomials overRis a ring with respect to the usual operations of

addition and multiplication of polynomials. Moreover, ifRis an integral domain thenR[x]is also an integral

domain.

Ideals

A subsetJof a ringRis called anidealinRif the following three properties hold:

(i) 0∈J.

(ii) For anya,b∈J, we havea−b∈J.

(iii) For anyr∈Randa∈J, we havera,ar∈J.