Question

8.
The union of two subrings
(a) will be a subring
(b)
(c) must be a subring if one is contained in the other​

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

Subring : A non empty subset S of a ring R is said to be a subring of R if S forms a ring under the binary operations of R .

• Let (R,+,•) be a ring , Then the non empty subset S of R is called a subring of R if (S,+,•) is a ring .

• A non empty subset S of a ring R is said be a subring of R iff for every a , b ∈ S → ab ∈ S and a - b ∈ S .

Solution :

To prove :

The union of two subrings will be a subring iff one of them is contained in another .

Proof :

(Please refer to the attachment)