Field & Integral Domain

Integral domain:-

1- A commutative ring with unity element and without zero divisors is called an integral domain.

2- An algebraic system (D,+,.) where D is a non-empty set with two binary compositions to be denoted by addition and multiplication is called an integral domain if following axioms are satisfied-

1- (D,+) is an abelian group.

2- (D,.) is an abelian semi-group with unity element.

3- Multiplication distribute addition.

4- If the product of two element is zero then at least one of them must be zero.

Field:-

1- A commutative ring with unity element in which every non-zero element has a multiplicative inverse is called a field.

2- An algebraic system (F,+,.) where F is a non-empty set with two binary compositions to be denoted bu addition and multiplication is called a field if following axioms are satisfied-

❶- (F,+) is an abelian group.

❷- (F-{0}, .) is an abelian group.

❸- Multiplication distribute addition.