Question

prove that the set Z of all integers is a ring w.r. t addition and multiplication of integers as the two ring composition​

Answer 1

Note :

Ring : A non empty set R equipped with two binary operations called addition and multiplication denoted by ( + ) and ( • ) is said to be a ring if the following properties holds :

1. (R,+) is an abelian group .
2. (R,•) is a semi-group
3. (R,+,•) holds distribute law .

• a•(b + c) = a•b + a•c
• (b + c)•a = b•a + c•a

Solution :

To prove :

(Z,+,•) is a ring

Proof :

1) Z is closed with respect to addition since the sum of two integers is again an integer .

ie. a + b ∈ Z for every a , b ∈ Z .

2) Addition of integers is associative .

ie. a + (b + c) = (a + b) + c for every a , b , c ∈ Z .

3) Addition of integers is commutative .

ie. a + b = b + a for every a , b ∈ Z .

4) Existence of additive identity :

ie. there exists 0 ∈ Z which is the additive identity in Z , since 0 + a = a + 0 for every a ∈ Z .

5) Existence of additive inverse :

ie. to each element a ∈ Z , there exists an element -a ∈ Z such that , -a + a = a + (-a) = 0 .

6) Z is closed with respect to multiplication .

ie. a•b ∈ Z for every a , b ∈ Z .

7) Multiplication is associative .

ie. a•(b•c) = (a•b)•c for every a , b , c ∈ Z .

8) Multiplication is distributive under addition .

ie. a•(b + c) = a•b + a•c and (b + c)•a = b•a + c•a for every a , b , c ∈ Z .

Hence , (Z,+,•) is a ring .