Axiom 1: Closure Under Addition

Closure under addition, to many who do not study mathematics, probably seem like the silliest thing to call an axiom. I can already hear you say, “well, addition always works, why do we need to specify that this is an axiom”. This is actually why I think Ring Theory is extremely cool. When we are discussing any of the axioms, we are looking at the very fundamentals of mathematics. We are looking at something so small, and yet so important, that most of us just take it for granted that addition will hold. In Ring Theory, we don’t take anything for granted.

So, what does it mean to have closure under addition? Well, in layman’s terms, we can say that closure under addition describes when adding to things together, they produce a sum that is also in the ring. In more technical terms, given any two elements a and b which are in the ring, a+b is also in the ring. This shows formally that addition holds in the ring.

To make it a bit clearer why this is vital to our definition of a ring, let me show you a number system that wouldn’t actually hold for this axiom, and thus disqualifies it as a ring. I want you to imagine the odd integers, so 1, 3, 5, -1, -3, -5, etc. Now, add them together. Do you get an odd number? I imagine you didn’t, and we have been taught from elementary school that two even number sum up to an even number, an even and odd number sum up to being odd, and two odd numbers sum up to being even. As such, the number system of only odd integers doesn’t actually hold under addition, which means that it cannot qualify as being a ring.

The last thing I would like to address under this is axiom is a natural follow-up question: “Does this mean that this axiom also covers subtraction?” Well, this axiom does not really guarantee subtraction. I think this is best shown by another example. Let’s say we are working with strictly positive integers, so 0, 1, 2, 3, etc. Now, this holds under our closure under addition axiom, since the addition of two positive integers is another positive integer. However, if you subtract a larger integer away from a smaller integer, say 4-5, then we aren’t in our number system anymore, which was the positive integers.

Another way to look at this is see how we translate subtraction into addition. As you’ll find, we only define rings on addition and multiplication. In this case, looking solely at addition, we can rewrite 4-5 as 4+(-5). Notice that when rewritten with addition, we are adding a negative integer, which wasn’t part of our system of only using positive integers. As such, it should hopefully make sense why this subtraction didn’t work, and it’s because we really weren’t working within the number system we had set for ourselves.

This video will give you another look at the Closure Property, mainly going over addition but also some other closure properties.