1. - K . A N I T H A M . S C . , M . P H I L . ,
- M . M O H A N A M A L A R M . S C . , M . P H I L . ,
III B.Sc Mathematics Sf
Modern Algebra-15UMTC51
2. Rings
A non empty set together with two binary operations
“+” and “.” satisfying the following axioms is called a
Ring.
i) (R,+) is an abelian group.
ii)”.” is an associative binary operation on R.
Iii)a.(b+c)=a.b+a.c and (a+b).c = a.c+b.c for all a,b,c
belongs to R.
3. Examples
(Z,+,.);(Q,+,.);(R,+,.);(C,+,.);(2z,+,.) are all rings.
{0} with binary operations + and . Defined as 0+0=0
and 0.0=0 is a ring.This is called null ring.
M2(R)under matrix addition and multiplication is a
ring.
4. Properties of Rings
0a=0
a(-b)=(-a)b=-(ab)
(-a)(-b)=ab
a(b-c)=ab-ac
A ring is called a Boolean ring if a2 = a
5. Isomorphism
Let (R,+,.) and (R/ ,+,.)be two rings. A bijection f from R
to R/ is called isomorphism if
i) f(a+b) = f(a) +f(b)
ii) f(ab) = f(a)f(b)
6. Types of Rings
A ring R is said to be commutative if ab=ba for all a,b
in R
A ring Ris said to be a ring with identity if there
exists an element 1 in R such that 1a=a1=a for all a in
R.
An element u in R is called a unit in R if it has a
multiplicative inverse in R
R is called a skew field if every non zero element in R
is a unit
7. Field
A commutative skew field is called a field.
Examples
Q,R and C are fields .
(Z,+,.)is not a field since 1 and -1 are the only
nonzero elements which have inverse.
8. Integral Domain
A nonzero element a in R is said to be a zero divisor
if there exists a nonzero element b in R such that
ab=0 or ba=0
A commutative ring with identity having no zero
divisors is called an integral domain.
Z is an integral domain
nZ is not an integral domain
Z7 is an integral domain
9. Results
A ring has no zero divisors iff cancellation law is
valid in R
Any unit in R cannot be a zero divisor
Z7 is an integral domain iff n is prime.
Any field F is an integral domain.
Any finite integral domain is a field.
Zn is a field iff n is prime.
10. Properties
R is commutative implies R/ is commutative
R is ring with identity implies R/ is ring with identity
R is an integral domain implies R/ is an integral
domain
R is a field implies R/ is a field
