Question

Prove that sum of two ideal is ideal

Answer 1

Step-by-step explanation:

Proposition The sum of any two ideals is an ideal. ... we see that the first term on the right J = I1 +···+Ik is an ideal and of course so is Ik+1, and so the sum J + I is an ideal. Thus the result holds for n = k + 1. So the result holds for all n ∈ {1,2,3, ...}.

Answer 2

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

Ideal : A non empty subset U of ring R is said to be an ideal (two sided ideal) of R if :

a , b ∈ U → a - b ∈ U and

a ∈ U , r ∈ R → ar ∈ U and ra ∈ U

Solution :

To prove :

Sum of two ideals of a ring is again an ideal .

Proof :

(Please refer to the attachment)