Question

Define characteristic of ring with example.

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

Boolean ring : (R,+,•) is said to be a boolean ring if x² = x for every x ∈ R .

Example : The ring {0 , 1} with respect to addition and multiplication forms a boolean ring .

Answer :

Characteristic of a ring , Ch(R) : If R is a ring , then any least positive integer n such that na = 0 for every n ∈ R , then n is called characteristic of R .

Example :

Let's find the characteristic of a boolean ring .

Let a ∈ R , then a² = a .

Now ,

→ (a + a)² = (a + a)

→ (a + a)(a + a) = a + a

→ a² + a² + a² + a² = a + a

→ a + a + a + a = a + a

→ a + a = 0

→ 2a = 0 for every a ∈ R

→ 2 is the characteristic of boolean ring R .

→ Ch(R) = 2