Question

a division ring is a ----- ring
(a)commotative (b)simple (c)unit (d) both b and c
​

Answer 1

Answer :

Simple

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

Division ring : If every non zero element of the ring R has multiplicative inverse in R , then R is called division ring .

Ideal : A non empty subset U of ring R is called an ideal (two slided ideal) of R if

1. a , b ∈ U → a - b ∈ U
2. a ∈ U , r ∈ R → ar ∈ U and ra ∈ U

Simple ring : A ring R is said to be simple ring if ;

1. there exist a , b ∈ R such that ab ≠ 0
2. R has no proper ideal , ie. only ideals of R are {0} and R .

Explanation :

Let R be a division ring and let A be any ideal of R such that A ≠ {0} .

Then there exists atleast one non-zero element in A , ie. a ∈ A , such that a ≠ 0 .

Since R is a division ring , thus

a ∈ R → a⁻¹ ∈ R

→ a•a^ = 1

Also , a ∈ A and a⁻¹ ∈ R

→ a•a⁻¹ ∈ A (by definition of ideal)

→ 1 ∈ A

Now , 1 ∈ A and r ∈ R

→ 1•r ∈ A (A is ideal)

→ r ∈ A

→ R ⊆ A ...........(1)

°•° A is ideal of R

→ A ⊆ R ...........(2)

From (1) and (2) , we have

A = R

→ A is trivial (improper) ideal

→ R has no proper ideal

→ R is a simple ring .

Hence , a division ring is a simple ring .