Question

The characteristic of the ring (Q,+,1) is

Answer 1

In mathematics, the characteristic of a ring R, often denoted char R is defined to be the smallest number of times one must use the ring's multiplicative identity 1 in a sum to get the additive identity 0.

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

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 .

Solution :

We know that , (Q,+,•) is a ring with unity 1 .

If 1 has infinite order , then there is no positive integer n such that n•1 = 0 .

Thus , Ch(Q) = 0

Now ,

Let's suppose that , 1 has additive order n . Then , n•1 = 0 and n is the least positive integer with this property .

Thus , for any x ∈ Q , we have

n•x = x + x + x + . . . + x (n summands)

= 1•x + 1•x + 1•x + . . . + 1•x (n summands)

= (1 + 1 + 1 + . . . + 1)•x (n summands)

= (n•1)•x

= 0•x (°•° n•1 = 0)

= 0

Hence ,

Characteristic of (Q,+,•) is n , ie. Ch(Q) = n .