Proving a Product of Commutative Rings not an Integral Domain:

[silvermane]

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 :)

 

[Dick]

I think that pretty much sums up why RXS isn't an integral domain.

 

[silvermane]

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