64 I The Expanding Universe of Numbers
A mapping f:R→R′of a ringRinto a ringR′is said to be a (ring)isomor-
phismif it is both bijective and a homomorphism. The inverse mappingf−^1 :R′→R
is then also an isomorphism. (Anautomorphismof a ringRis an isomorphism ofR
with itself.)
Thus we have shown that, if f:R→R′is a homomorphism of a ringRinto a
ringR′, with kernelN, then the quotient ringR/Nis isomorphic tof(R).
An idealMof a ringRis said to bemaximalifM=Rand if there are no ideals
Ssuch thatM⊂S⊂R.
LetMbe an ideal of the ringR.IfSis an ideal ofRwhich containsM, then the
setS′of all cosetsM+awitha∈Sis an ideal ofR/M.Conversely,ifS′is an ideal
ofR/M, then the setSof alla∈ Rsuch thatM+a∈S′is an ideal ofRwhich
containsM. It follows thatMis a maximal ideal ofRif and only ifR/Mis simple.
Hence an idealMof a commutative ringRis maximal if and only if the quotient ring
R/Mis a field.
To conclude, we mention a simple way of creating new rings from given ones. Let
R,R′be rings and letR×R′be the set of all ordered pairs (a,a′) witha∈Rand
a′∈R′. As we saw in the previous section,R×R′acquires the structure of a (com-
mutative) group under addition if we define the sum(a,a′)+(b,b′)of(a,a′)and
(b,b′)to be (a+b,a′+b′). If we define their product(a,a′)·(b,b′)to be(ab,a′b′),
thenR×R′becomes a ring, with( 0 , 0 ′)as identity element for addition and( 1 , 1 ′)as
identity element for multiplication. The ring thus constructed is called thedirect sum
ofRandR′, and is denoted byR⊕R′.
9 Vector Spaces and Associative Algebras
Although we assume some knowledge of linear algebra, it may be useful to place the
basic definitions and results in the context of the preceding sections. A setVis said
to be avector spaceover a division ringDif it is a commutative group under an
operation+(addition)and there exists a mapφ:D×V→V(multiplication by a
scalar) such that, ifφ(α,v)is denoted byαvthen, for allα,β∈Dand allv,w∈V,
(i)α(v+w)=αv+αw,
(ii)(α+β)v=αv+βv,
(iii)(αβ)v=α(βv),
(iv) 1v=v,
where 1 is the identity element for multiplication inD. The elements ofVwill be
calledvectorsand the elements ofDscalars.
For example, for any positive integern,thesetDnof alln-tuples of elements of
the division ringDis a vector space overDif addition and multiplication by a scalar
are defined by
(α 1 ,...,αn)+(β 1 ,...,βn)=(α 1 +β 1 ,...,αn+βn),
α(α 1 ,...,αn)=(αα 1 ,...,ααn).The special casesD=RandD=Chave many applications.