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

APP. B] ALGEBRAIC SYSTEMS 445

Note first thatJis a subring ofR. Also,Jis a subgroup (necessarily normal) of the additive group ofR. Thus
we can form the following collection of cosets which form a partition ofR:

{a+J|a∈R}
The importance of ideals comes from the following theorem which is analogous to Theorem B.7 for normal
subgroups.

Theorem B.10: LetJbe an ideal in a ringR. Then the cosets{a+J|a∈R}form a ring under the coset
operations

(a+J)+(b+J)=a+b+J and (a+J )(b+J)=ab+J

This ring is denoted byR/Jand is called thequotient ring.

Now letRbe a commutative ring with an identity element 1. For anya∈R, the following set is an ideal:

(a)={ra|r∈R}=aR

It is called theprincipal ideal generatedbya. If every ideal inRis a principal ideal, thenRis called aprincipal
ideal ring. In particular, ifRis also an integral domain, thenRis called aprincipal ideal domain(PID).

EXAMPLEB.14

(a) Consider the ringZof integers. Then every idealJinZis a principal ideal, that is,J=(m)=mZ,for

some integerm. ThusZis a principal ideal domain (PID). The quotient ringZm=Z/(m)is simply the ring

of integers modulom. AlthoughZis an integral domain (no zero divisors), the quotient ringZmmay have

zero divisors, e.g., 2 and 3 are zero divisors inZ 6.

(b) LetRbe any ring. Then {0} andRare ideals. In particular, ifRis a field, then {0} andRare the only ideals.

(c) LetKbe a field. Then the ringK[x]of polynomials overKis a PID (principal ideal domain). On the other

hand, the ringK[x, y]of polynomials in two variables is not a PID.

Ring Homomorphisms
A mappingffrom a ringRinto a ringR′is called aring homomorphismor, simply,homomorphismif,
for everya,b∈R,
f(a+b)=f(a)+f (b), f (ab)=f(a)f(b)

In addition, iffis one-to-one and onto, thenfis called anisomorphism; andRandR′are said to beisomorphic,
writtenR∼=R′.
Supposef:R→R′is a homomorphism. Then the kernel off, written Kerf, is the set of elements
whose image is the zero element 0 ofR′; that is,

Kerf={r∈R|f(r)= 0 }

The following theorem (analogous to Theorem B.9 for groups) is fundamental to ring theory.

Theorem B.11: Letf:R→R′be a ring homomorphism with kernelK. ThenKis an ideal inR, and the
quotient ringR/Kis isomorphic tof(R).

Divisibility in Integral Domains

Now letDbe an integral domain. We say thatbdividesainDifa=bcfor somec∈D. An elementu∈D

is called aunitifudivides 1, i.e., ifuhas a multiplicative inverse. An elementb∈Dis called anassociateof

a∈Difb=uafor some unitu∈D. A nonunitp∈Dis said to beirreducibleifp=abimpliesaorbis a

unit.

An integral domainDis called aunique factorization domain(UFD), if every nonunita∈Dcan be written

uniquely (up to associates and order) as a product of irreducible elements.