Question

Characteristic of integral domain i zero or prime

Answer 1

The characteristic of an integral domain is (or prime). My lecture has not yet covered infinite integral domain but I'll like to understand the proof. Basic fact: is an integral domain so is a commutative ring with unity (multiplicative inverse = exists) containing no zero-divisor.

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 .

Integral domain : A commutative ring R with unity is called a integral domain if it has no zero divisor . ie. R is an integral domain a•b , a , b ∈ R → atleast one of a or b is zero .

Solution :

To prove :

The characteristic of an integral domain R is either 0 or prime .

Proof :

If there does not exist any least positive integer n such that n•a = 0 for every a ∈ R , then R is of characteristic 0 .

But if there exists such a least positive integer , then let it be m . Thus m•a = 0 for every a ∈ R and particularly m•1 = 0 for a = 1 .

If m is not prime , then there are integers m₁ and m₂ such that , m > m₁ , m₂ > 0 and m = m₁m₂ .

Since , m•1 = 0

→ (m₁m₂)•1 = 0

→ (m₁•1)(m₂•1) = 0

and since R is Integral domain therefore either m₁•1 = 0 or m₂•1 = 0 which is a contradiction as m is the least positive integer such that m•1 = 0 .

Thus , m must be prime .

Hence proved .