Axiom 11: Integral Domain

After having studied all these axioms extensively, I feel like most people would know what these axioms are. Especially if you’re still in high school or middle school, you might even be studying these terms and ideas in class for the first time. However, I most certainly didn’t know what an Integral Domain was my middle high school years, and I certainly didn’t know what was required to achieve that title.

This time, I’m going to assume you don’t know what an Integral Domain is even remotely, let alone the proper definition. So, starting at that point, what in the world is an integral domain, why does it sound like it has something to do with integers, and what does it take to create? First of all, an integral domain requires the knowledge of what a zero divisor is. A zero divisor is, quite simply put, a number that isn’t the additive identity, when multiplied by another number that isn’t the additive identity, gives us the additive identity. In terms of the real numbers, since it’s easier to visualize, when you multiply two numbers that aren’t zero, you still get zero. Now, that doesn’t make sense in the real numbers, and I think that intuitively makes sense with our knowledge of the real numbers. However, with other number systems, this isn’t nearly as clear. That’s why I stress that this number is the additive identity, not zero. In other systems, the additive identity isn’t zero, and thus, when looking at other Rings and Integral Domains, we shouldn’t be making the extremely poor assumption that the additive identity will be zero.

Now that we know what a zero divisor is, what is the definition of an Integral Domain? In the following definition, please make note of the fact that I will use “1” when referring to the multiplicative identity element and “0” when referring to the additive identity element. So, the definition of an Integral Domain is: An integral domain is a commutative ring R with identity “1” not equal to “0” that satisfies the following “whenever a and b in R and ab=”0”, then a=”0” or b=”0”.” If you’ve even glanced through the rest of the site, you will notice that this definition is by far the most complex, with the most parts to discuss. In English, if two numbers multiply to equal “0,” then one of those numbers was “0”. Since I did spend a paragraph discussing zero divisors, this definition basically says that there are no zero divisors in the number system.

An Integral Domain has a lot of prerequisites. First, it must both be a commutative ring and a ring with identity, which means axioms 9 and 10 must hold on top of the first basic 8. Also, we explicitly state that “1” and “0” can’t be the same, otherwise, we could have a ring that is made up of just “0”. This type of ring, at least to me, is frankly trivial and very boring, so it’s excluded.

I hope this gives you more insight into an Integral Domain. This is easily the most complex axiom to wrap your head around, so if you’ve followed this, then congratulations! If you’ve gotten this far, move on to the final axiom that defines a Field!


This video gives you an in-depth look at Integral Domains, including an example of something that isn't an Integral Domain. Do be aware that if you do not know about modular arithmetic, this video may be somewhat complicated to follow. Think about it like clocks. 11+3=2 in clock arithmatic, right? This is because we only go to 12! This is basically how modular arithmatic works, so think of it like that when following the video.