when does 1 ever equal zero? integral domain question 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
 A: 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."
A: It may help to realise that what we are talking about here is abstract algebra.  This means that we are concentrating on the properties satisfied by "objects" and "operations" rather than the specific details of those objects and operations.
So here is an example: I am going to talk about the number $0$ and no other numbers at all, ever!  (Very boring? - yes definitely.)  Which of the axioms of an integral domain does this satisfy.  Well for example, it satisfies the associative law since the only possible sums of three numbers we have are
$$0+(0+0)\quad\hbox{and}\quad (0+0)+0\ ,$$
and these are definitely equal.  Another: is there a number $u$ such that $ux$ is always equal to $x$?  Well, the only possible value of $x$ is $0$ - that's the only number we're talking about, remember?  So all we need is $u0=0$, and that's true when $u=0$.  So $0$ is the multiplicative identity, and we usually write the multiplicative identity as $1$.  Remember, that doesn't mean it is $1$ in the usual sense - as I said above, we are not really paying attention to the specific details of our numbers.  So in this case it is reasonable to say that $0=1$.
But as mentioned above, this is a really boring example and is not typical of other integral domains.  That's why we choose to specifically exclude it in the definition.
A: The detail that $1\neq0$ is typically stated as "an integral domain is a nonzero commutative ring..." by most accounts. Indeed, there is exactly one ring with this property: http://en.wikipedia.org/wiki/Zero_ring
