I'm getting frustrated with the definition of an integral domain. I'm trying to prove the Gaussian integers are an integral domain, having just proven they're a subring of the complex numbers.
However, I understand the part about it being a commutative ring where it has no zero divisors, but what on Earth does it mean to say 1 can't equal 0… When is that ever the case!?
I got this definition from the web:
An integral domain is a commutative ring with an identity (1 =\= 0) with no zero-divisors.
That is ab = 0 implies a = 0 or b = 0.
So please explain to me carefully what they mean
Thanks
Best Answer
Note that 1 is the multiplicative identity and 0 times anything is 0. Thus if $1=0$ we have $0x=0=x=1x$ for all $x$, so the ring has exactly one element, and that is 0.
Even though this satisfies the other conditions of an integral domain because if $ab=0$ then $a=0$ and $b=0$, it is explicitly excluded from the definition.
When we say "1" and "0" in ring theory, we are not speaking of numbers necessarily. It is somewhat an abuse of notation. "1" means "the multiplicative identity of the ring we are considering" and "0" refers to the additive identity. You are right in saying that the numbers 1 and 0 are not equal. It may be more comfortable for you to read "1=0" as "the multiplicative identity is equal to the additive identity."