Prove the set of integers is a commutative ring with identity

In summary, the conversation discusses the process of proving that the integers form a commutative ring. It is suggested that only the commutative multiplication property needs to be proven, but this is already a known property of the set of integers. The same argument applies to the identity element. Ultimately, it is recommended to consult with a professor for guidance on whether these facts can be cited without explicit proof.
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How should one prove that the integers form a commutative ring? I am not sure exactly where to go with this and how much should be explicitly shown.

A ring is meant to be a system that shares properties of Z and Zn. A commutative ring is a ring, with the commutative multiplication property. Only need to prove then that that the integers have a commutative multiplication property in that case?? But commutativity of multiplication is a known property of the set of integers, "an arithmetical fact" as my book says. So do I just cite this fact/theorem without having to show much algebra bingo bango its a com. ring? Same argument witht the identity elemnt. It is part of the list of arithmetic facts given to us, which themselves are not proven, so just cite it?
 
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  • #2
We can't answer this. You need to ask your professor whether you can just cite these facts without explicitly showing them.
 
  • #3
I had that feeling haha. Thanks.
 

1. 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 satisfy certain properties. These properties include commutativity, associativity, distributivity, and the existence of an identity element.

2. How do you prove that a set of integers is a commutative ring?

To prove that a set of integers is a commutative ring, we need to show that it satisfies all the properties of a commutative ring. This includes proving that addition and multiplication are commutative and associative, that multiplication is distributive over addition, and that the set contains an identity element for both operations.

3. What does it mean for a ring to have an identity element?

An identity element is an element in a ring that, when combined with any other element using a specific operation (such as addition or multiplication), results in the other element. In other words, the identity element does not change the value of the other element when combined with it.

4. Can you give an example of an element in the set of integers that does not have an inverse?

Yes, 0 is an element in the set of integers that does not have an inverse. This is because there is no integer that, when added to 0, results in 1 (the identity element for addition).

5. Why is it important to prove that a set of integers is a commutative ring?

Proving that a set of integers is a commutative ring is important because it allows us to use the properties of rings to solve mathematical problems involving integers. It also provides a solid foundation for more complex mathematical concepts that build upon the concept of a ring, such as fields and vector spaces.

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