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

APP. B] ALGEBRAIC SYSTEMS 443

Theorem B.9: Supposef:G→G′is a homomorphism with kernelK. ThenKis a normal subgroup ofG, and
the quotient groupG/Kis isomorphic tof (G).

EXAMPLE B.12

(a) LetGbe the group of real numbers under addition, and letG′be the group of positive real numbers under

multiplication. The mappingf:G→G′defined byf(a)= 2 ais a homomorphism because

f(a+b)= 2 a+b= 2 a 2 b=f(a)f(b)

In fact,fis also one-to-one and onto; henceGandG′are isomorphic.

(b) Letabe any element in a groupG. The functionf:Z→Gdefined byf (n)=anis a homomorphism since

f(m+n)=am+n=am·an=f (m)·f (n)

The image offisgp(a), the cyclic subgroup generated bya. By Theorem B.9,

gp(a)∼=Z/K

whereKis the kernel off.IfK={ 0 }, thengp(a)=Z. On the other hand, ifmis the order ofa, then

K={multiples ofm}, and sogp(a)∼=Zm. In other words, any cyclic group is isomorphic to either the

integersZunder addition, or toZm, the integers under addition modulom.

B.6 RINGS, INTEGRAL DOMAINS, AND FIELDS

LetRbe a nonempty set with two binary operations, an operation of addition (denoted by+) and an operation
of multiplication (denoted by juxtaposition). ThenRis called aringif the following axioms are satisfied:

[R 1 ] For anya,b,c∈R, we have(a+b)+c=a+(b+c).

[R 2 ] There exists an element 0∈R, called thezeroelement, such that, for everya∈R,

a+ 0 = 0 +a=a.

[R 3 ] For eacha∈Rthere exists an element−a∈R, called thenegativeofa, such that

a+(−a)=(−a)+a= 0.

[R 4 ] For anya,b∈R, we havea+b=b+a.

[R 5 ] For anya,b,c∈R, we have(ab)c=a(bc).

[R 6 ] For anya,b,c∈R, we have: (i)a(b+c)=ab+ac, and (ii)(b+c)a=ba+ca.

Observe that the axioms [R 1 ] through [R 4 ] may be summarized by saying thatRis an abelian group under
addition.
Subtraction is defined inRbya−b=a+(−b).
One can prove (Problem B.21) thata· 0 = 0 ·a=0 for everya∈R.
A subsetSofRis asubringofRifSitself is a ring under the operations inR. We note thatSis a subring of
Rif: (i) 0∈S, and (ii) for anya,b∈S, we havea−b∈Sandab∈S.