Saturday, May 20, 2006

Fields and Rings

In today's blog, I will talk about the assumptions behind the idea of a field. I use the concept of a field in the proof for the Division Algorithm for Polynomials.

Definition 1: Ring

A ring R is a nonempty set with two binary operations: addition (a + b) and multiplication (ab).

It has the following properties:

1. Commutative Rule for Addition: a + b = b + a

2. Associative Rule for Addition: (a + b) + c = a + (b + c)

3. Additive Identity: there exists a value 0 ∈ R such that a + 0 = a for all a ∈ R

4. Additive Inverse: for all elements a ∈ R, there exists a value -a ∈ R such that a + -a = 0.

5. Associative Rule for Multiplication: a(bc) = (ab)c

6. Distributive Rule: a(b + c) = ab + ac and (b + c)a = ba + ca

Definition 2: Commutative Ring

A ring that has the following additional property:

7. Commutative Rule for Multiplication: ab = ba

Definition 3: Field

A field is a commutative ring R with unity in which every nonzero element is a unit.

It has all the properties of a commutative ring plus:

8. Multiplicative Identity (unity): there exists a value 1 ∈ R such that such that a * 1 = a for all a ∈ R

9. Multiplicative Inverse (every nonzero element is a unit): for all nonzero elements a ∈ R, there exists an inverse a-1 ∈ R such that a*a-1 = 1.

Examples:

1. The set of integers Z is a commutative ring with unity 1. [See here for more information on the integers]

2. The set Zn is a commutative ring with unity 1. [See here for more information on modular arithmetic]

3. The set of 2x2 matrices with integer entries is a noncommutative ring with unity. [See here for more information on 2x2 matrices]

4. If p is prime, then the set Zp is a field. [See here for more information on modular arithmetic]

5. The set of rational numbers Q is a field.

6. The set of real numbers R is a field.

7. The set of complex numbers is a field. [See here for more information on complex numbers]

8. The set of Gaussian Integers is a ring. [See here for more information on Gaussian Integers]

9. The set of Eisenstein Integers is a ring. [See here for more information on Eisenstein Integers]

References

5 comments:

Anonymous said...

Dear, sir, my english is not as good as I would like. So, sorry for mistakes. This is not related with rings and fields, but as I have read several of your writtings, I supose you can give me an ansewr.
I am now engaged in studing numerical series, and I have met one that I can not solve. This is the sum from 1 to infinite of !/(n·n!)
May you help me?
I know that the sum is convergent according to D'Alembert criteria, but I do not find the way for getting the sum.
My mail is fjarosa@gmail.com

Larry Freeman said...

Hi Fjarosa,

I haven't spent too much time on factorial sums.

I would start here

Since we know that sum(1/n!) = e - 1, it might be helpful to view 1/n*n! as

1/n*n! = (1/1 + ... + 1/n!) + (1/2 + ... + 1/n!) + (1/6 + ... + 1/n!) + ... + (1/n!)


Good luck

-Larry

Anonymous said...

Thank for your usefull help I have tu study all this material
Javier

bloomy said...

Dear Sir,my question is that, 1)Is an infinite ring has a fininte subring?
2)If yes,then that finite subring is a subgroup of infinite ring becausethe ring is itself group and according to Lagrange theorem for groups i.e The order of a subgroup divides the order of the group but here the order of the grop is infinite and the order of the subgroup is finite?


Sorry for mistakes in English.
My mail is bloomy_boy2006@yahoo.com

bloomy said...

Dear Sir hope u r fine.My questions are that

1)Is an infinite ring has a finite subring?
2)If your aanswer is yes, then that finite subring must the subgroup of the infinite ring because the ring itself is a group then accordind to Lagrang's theorem order of the subgroup divides the order of the group but here the order of the group is infinite and the order of the subgroup is finite?