Proving a Product of Commutative Rings not an Integral Domain:

In summary, the conversation discusses proving that the product of two commutative rings, R X S, cannot be an integral domain. It is shown that R X S is not an integral domain because it contains the element (0,0) which is a zero-divisor. The conversation concludes that this is a logical proof and no further explanation is needed.
  • #1
silvermane
Gold Member
117
0

Homework Statement


Prove that if we have two commutative rings R and S and form the product R X S, then R X S cannot be an integral domain.


The Attempt at a Solution


We have that an integral domain is a commutative ring with 1 not= 0 and with non-zero zero-divisors.

==> (1,0)X(0,1) = (0,0), Thus it's not an integral domain.

I just want to make sure I'm doing this logically correct :)
 
Physics news on Phys.org
  • #2
I think that pretty much sums up why RXS isn't an integral domain.
 
  • #3
Dick said:
I think that pretty much sums up why RXS isn't an integral domain.

lol that's what I thought, just wasn't sure if i needed more!
It seems too easy... :P
 

FAQ: Proving a Product of Commutative Rings not an Integral Domain:

1. What is a commutative ring?

A commutative ring is a mathematical structure consisting of a set of elements, along with two operations called addition and multiplication. These operations must follow certain rules, including the commutative property, where the order of operands does not affect the result.

2. What is an integral domain?

An integral domain is a type of commutative ring where the multiplication operation also satisfies the cancellation property. This means that if the product of two elements is equal to zero, then at least one of the elements must be equal to zero.

3. How can you prove that a product of commutative rings is not an integral domain?

To prove that a product of commutative rings is not an integral domain, you can show that the cancellation property does not hold. This can be done by finding two non-zero elements whose product is equal to zero, or by showing that the product of two non-zero elements is not equal to zero.

4. Can a product of commutative rings be both an integral domain and not an integral domain?

No, a product of commutative rings cannot be both an integral domain and not an integral domain at the same time. It can either satisfy the properties of an integral domain or not, but not both simultaneously.

5. What is the significance of proving that a product of commutative rings is not an integral domain?

Proving that a product of commutative rings is not an integral domain can help us understand the structure of the rings better. It can also help us identify the properties that are not satisfied, which can lead to further research and discoveries in the field of mathematics.

Similar threads

Showing that a ring is an integral domain
Replies
10
Views
2K
Abstract Algebra: Ring Proof (Multiplicative Inverse)
Replies
1
Views
2K
Nontrivial Finite Rings with No Zero Divisors
Replies
3
Views
2K
Rings and Fields- how to prove Z[x]/pZ[x] is an integral domain
Replies
1
Views
2K
Calculate this Integral around the Circular Path using Green's Theorem
Replies
3
Views
1K
Showing isomorphism between fractions and a quotient ring
Replies
1
Views
1K
Integral Domain, r^2 = r proof that r = 0 or 1
Replies
5
Views
1K
Finite Commutative Ring: Proving Integral Domain w/ No Zero Divisors
Replies
4
Views
2K
Factor Ring of a Ring: Example of Integral Domain with Divisors of 0
Replies
3
Views
2K
Can Zero Divisors in Polynomial Rings Be Characterized?
Replies
13
Views
2K

Hot Threads

Recent Insights

Back
Top