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Simply typing this question into google, I get:

An integral domain is a commutative ring with an identity (1 0) with no zero-divisors. That is ab = 0 a = 0 or b = 0.

I don't understand what they mean by an 'identity (1 0)' with no 'zero-divisors'

What exactly is an integral domain then in layman's terms

I'm currently trying to show Z{i} (the Gaussian integers) is an integral domain, and I've just shown Z{i} is a subring of C. So any help specific to this would be greatly appreciated,

Thanks!

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  • $\begingroup$ Look at the page Google links. It says $1\neq 0$, i.e., the (multiplicative) identity cannot be the additive one. The Gaussian integers are NOT a subring of the integers. $\endgroup$
    – TorsionSquid
    Feb 18, 2015 at 0:19
  • $\begingroup$ It appears you are reading this page and your browser is not rendering the unequal sign. It should be "with an identity $\,(1\ne 0)\ $ ..." $\endgroup$
    – Bill Dubuque
    Feb 18, 2015 at 0:19
  • $\begingroup$ The $(1 0)$ part is just a Google mistranslation. It should be $1 \neq 0$. And definition you wrote says exactly what having no zero-divisors means: that if $ab = 0$, then either $a = 0$ or $b = 0$. $\endgroup$
    – Christopher A. Wong
    Feb 18, 2015 at 0:19
  • $\begingroup$ How exactly is $\Bbb Z[i]$ a subring of $\Bbb Z$? $\endgroup$
    – Tim Raczkowski
    Feb 18, 2015 at 0:20
  • $\begingroup$ I'm reading, and I'm not sure what the notation Z6 means... it goes on to say 2.3=0 if that helps. I've seen it before but can't remember $\endgroup$
    – Douglas
    Feb 18, 2015 at 0:22

1 Answer 1

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A zero divisor is a number that divides zero. More specifically, an element $a$ in a ring $R$ is a zero divisor if there exists some other element $b\neq 0$ such that $ab = 0$. For instance, in the ring of integers mod $4$, the number $2$ is a zero divisor since $2\cdot 2 \equiv 0$.

An integral domain is a ring without zero divisors.

A field is an integral domain where every nonzero element has an inverse.

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