8 Rings and Fields 63In a ring with no divisors of zero, the additive order of any nonzero elementais
the same as the additive order of 1, sincema=(m 1 )a=0 if and only ifm 1 =0.
Furthermore, the characteristic of such a ring is either 0 or a prime number. For assume
n=lm,wherelandmare positive integers less thann.Ifn 1 =0, then
(l 1 )(m 1 )=n 1 = 0.Since there are no divisors of zero, eitherl 1 =0orm 1 =0, and hence the character-
istic cannot ben.
A subsetSof a ringRis said to be a (two-sided)idealif it is a subgroup ofR
under addition and if, for everya∈Sandc∈R, bothac∈Sandca∈S.
Any ringRhas two obvious ideals, namelyRitself and the subset{ 0 }.Itissaidto
besimpleif it has no other ideals and is nontrivial.
Any division ring is simple. For if an idealSof a division ringRcontainsa=0,
then for everyc∈Rwe havec=(ca−^1 )a∈S.
Conversely, if acommutativeringRis simple, then it is a field. For, ifais any
nonzero element ofR,theset
Sa={xa:x∈R}is an ideal (sinceRis commutative). SinceSacontains 1a=a=0, we must have
Sa=R. Hence 1=xafor somex∈R. Thus every nonzero element ofRis invertible.
IfRis a commutative ring anda 1 ,...,am ∈R, then the setSconsisting of all
elementsx 1 a 1 +···+xmam,wherexj∈R( 1 ≤j≤m), is clearly an ideal ofR,the
idealgeneratedbya 1 ,...,am. An ideal of this type is said to befinitely generated.
We now show that ifSis an ideal of the ringR, then the setSof all cosetsS+a
ofScan be given the structure of a ring. The ringRis a commutative group under
addition. Hence, as we saw in§7,Sacquires the structure of a (commutative) group
under addition if we define the sum ofS+aandS+bto beS+(a+b).Ifx=s+aand
x′=s′+bfor somes,s′∈S,thenxx′=s′′+ab,wheres′′=ss′+as′+sb.Since
Sis an ideal,s′′∈S. Thus without ambiguity we may define the product of the cosets
S+aandS+bto be the cosetS+ab. Evidently multiplication is associative,S+ 1
is an identity element for multiplication and both distributive laws hold. The new ring
thus constructed is called thequotient ringofRby the idealS, and is denoted byR/S.
A mappingf:R→R′of a ringRinto a ringR′is said to be a (ring)homomor-
phismif, for alla,b∈R,
f(a+b)=f(a)+f(b), f(ab)=f(a)f(b),and iff( 1 )= 1 ′is the identity element for multiplication inR′.
Thekernelof the homomorphismfis the setNof alla∈Rsuch thatf(a)= 0 ′
is the identity element for addition inR′. The kernel is an ideal ofR, since it is a
subgroup under addition and sincea∈N,c∈Rimplyac∈Nandca∈N.
For anya∈R, puta′=f(a)∈R′.ThecosetN+ais the set of allx∈Rsuch
thatf(x)=a′,andthemapN+a→a′is a bijection from the collection of all cosets
ofNtof(R).Sincefis a homomorphism,N+(a+b)is mapped toa′+b′and
N+abis mapped toa′b′. Hence the mapN+a→a′is also a homomorphism of the
quotient ringR/Nintof(R).