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integral domain

 

Def .

Integral Domain :

A commutative ring with unity and without zero divisors is called an integral  domain

Results :

 If R is a ring then for all a, b, c  ÎR , we have

1. a+ b = b +c   Þ b = c

2. (-a)= - a

3. The zero element of R is unique.

 4. The additive inverse of any element in R is unique.

5.a.0=0.a=0

6. a(-b)= -(ab)

7. Multiplicative inverse of a non-zero element, if exists, is unique.

Key points:

1.       A commutative ring R with unity is an integral domain iff cancellation laws holds in R.

2.        A division ring is always without zero divisors.

3.        A field is always without zero divisors.

4.        A field is always an integral domain.

5.        A ring with zero divisors cannot be an integral domain and field.

6.       Wedderburn’s theorem :

(i)                  Every finite commutative ring without zero divisors is a field.

(ii)                Every finite integral domain is a field. But an infinite integral domain may or may not be a field.

 Boolean Ring :

 A ring (R,+,.) in which every element is an idempotent is called a Boolean ring.

 

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