Proving Z is a Ring with Addition and Subtraction

In summary, the conversation was about proving that Z with a specific addition and subtraction operation is a ring. The person had already proven the axioms for addition but was stuck on the multiplication part. They provided two equations and asked how they were equal. After expanding the equations, it became clear that they were indeed equal and the person realized their mistake.
  • #1
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Homework Statement


Prove that Z with the following addition and subtraction is a ring.



Homework Equations


a[tex]\oplus[/tex]b = a + b - 1 and a[tex]\odot[/tex]b = ab - (a + b) + 2



The Attempt at a Solution

I proved all the axioms for addition. I'm stuck on the multiplication part.

(a[tex]\odot[/tex]b)[tex]\odot[/tex]c = (ab-(a+b)+2)c - (ab-(a+b)+2+c) + 2

a[tex]\odot[/tex](b[tex]\odot[/tex]c) = a(bc-(b+c)+2) - (a+bc-(b+c)+2) + 2

How are these equal? I know it's a ring because a couple problems later, my books wants me to prove that it's an integral domain...
 
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  • #2
I think they are equal. Just expand them out.
 
  • #3
(ab-(a+b)+2)c - (ab-(a+b)+2+c) + 2 =

abc-ac-bc+2c-ab+a+b-2-c+2 = abc-ac-bc-ab+c+a+b

a(bc-(b+c)+2) - (a+bc-(b+c)+2) + 2 =

abc-ab-ac+2a-a-bc+b+c-2+2 = abc-ab-ac-bc+a+b+c

It looks so much clearer now. My own handwriting was deceiving me... Gah, that's the second stupid question I've posted this weekend...
 

Related to Proving Z is a Ring with Addition and Subtraction

What is a ring in mathematics?

A ring in mathematics is a mathematical structure that consists of a set of elements and two binary operations - addition and multiplication. The set must follow certain properties, such as closure, associativity, distributivity, and the existence of an identity element.

How do you prove that Z is a ring with addition and subtraction?

To prove that Z (the set of integers) is a ring with addition and subtraction, we need to show that it satisfies all the properties of a ring. These properties include closure, associativity, commutativity, distributivity, and the existence of an identity and inverse element.

What is the difference between a ring and a field?

A field is a more specialized type of ring that also includes the operation of division. In a field, every non-zero element has a multiplicative inverse, while in a ring, this is not always the case. Additionally, a field must also satisfy the commutative property for multiplication, while a ring does not necessarily have to.

Can you give an example of a non-commutative ring?

One example of a non-commutative ring is the set of n×n matrices with entries in the real numbers, where n is greater than or equal to 2. In this ring, the order of multiplication matters, so it is not commutative.

Why is it important to prove that Z is a ring with addition and subtraction?

Proving that Z is a ring with addition and subtraction is important because it helps us understand the properties and structure of this set. It also allows us to make connections and apply concepts from ring theory to other areas of mathematics, such as algebra and number theory.

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