Commutative Ring Definition: What is a Commutative Ring?

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In summary: If you intended "regular" subtraction, this is not a ring because it does not have an "associative inverse" or a "negative number". There are a number of axioms that a ring must satisfy for it to be called a ring, including the axiom of addition, which is that for every two integers a,b in A a+b = a. This set, A, does not satisfy this axiom because 0 is not in A.
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Miike012
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Commutative Ring: Let R be a set of elements a,b,c ... for which the sum (a + b) and the product (ab) of any two elements a and b of R are defined is called a commutative ring

This is my understanding, tell me if I am wrong...

f(x) = 1/x
Domain of f: { x | x =/ 0 } = A

Thus any two integers (a + b) = 0 and (ab) = 0 are not in the commutative ring... commutative ring being A
Is this correct?
 
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Why are you saying that for every two integers a,b in A a+b = 0 holds?

This set is not even a ring (let alone a commutative ring) because it fails to hold a certain axiom. Any guesses which one? Can you prove it?
 
  • #3
I just opened the book.. Never took a class on this... so I am confused ab out why this definition is important.
And I was hoping for an example for the definition...

It also goes on to say that it must follow 8 rules
 
  • #4
http://en.wikipedia.org/wiki/Ring_(mathematics )
 
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Miike012 said:
Commutative Ring: Let R be a set of elements a,b,c ... for which the sum (a + b) and the product (ab) of any two elements a and b of R are defined is called a commutative ring

This is my understanding, tell me if I am wrong...

f(x) = 1/x
Domain of f: { x | x =/ 0 } = A

Thus any two integers (a + b) = 0 and (ab) = 0 are not in the commutative ring... commutative ring being A
Is this correct?
A "ring" consists of a set of objects together with two operations, * and +, satisfying a lot of requirements- mostly those for the integers with regular addition and subtraction. You have defined your set, A, the set of all nonnegative numbers, but have not defined addition or subtraction. If you intended "regular" addition of numbers, this is not a ring because it does not have an "additive identity"- you have specifically excluded 0 which is the identity for "regular" addition.
 

FAQ: Commutative Ring Definition: What is a Commutative Ring?

What is a commutative ring?

A commutative ring is a mathematical structure that consists of a set of elements, along with two binary operations (usually addition and multiplication) that follow certain rules, such as associativity, commutativity, and distributivity.

What are the properties of a commutative ring?

A commutative ring must have the following properties:

  • Commutativity: The order of operations does not matter. For example, a*b = b*a.
  • Associativity: The way elements are grouped in an operation does not matter. For example, (a+b)+c = a+(b+c).
  • Identity elements: There must be an additive identity (usually 0) and a multiplicative identity (usually 1).
  • Inverses: Every element must have an additive inverse (an element that, when added to it, gives the additive identity) and a multiplicative inverse (an element that, when multiplied by it, gives the multiplicative identity).
  • Distributivity: Multiplication must distribute over addition. For example, a*(b+c) = a*b + a*c.

Are all rings commutative?

No, not all rings are commutative. A ring can be commutative or non-commutative, depending on whether the multiplication operation is commutative or not.

What is an example of a commutative ring?

An example of a commutative ring is the set of integers (Z) with addition and multiplication as the binary operations. In this case, the commutativity property holds, as well as all the other properties of a commutative ring.

What is the difference between a commutative ring and a field?

A field is a type of commutative ring that also has the property of multiplicative inverses for all non-zero elements. In other words, every non-zero element in a field has a multiplicative inverse, while in a commutative ring, this may not always be the case. Additionally, a field must have a finite number of elements, while a commutative ring can have an infinite number of elements.

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