Question

what is the ring and property of it of It in Algebra .​

Answer 1

Answer :

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 closed with respect to addition .

ie. a + b ∈ R for every a , b ∈ R .

2) Addition is associative .

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

3) Addition is commutative .

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

4) Existence of additive identity :

ie. there exists an additive identity in R denoted by 0 such that , 0 + a = a + 0 for every a ∈ R .

5) Existence of additive inverse :

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

6) R is closed with respect to multiplication .

ie. a•b ∈ R for every a , b ∈ R .

7) Multiplication is associative .

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

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 ∈ R .

In other words , an algebraic structure (R,+,•) is said to be a ring if ;

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