Axiom 4: An Additive Identity

The Axiom for an Additive Identity could easily be one of the more confusing axioms that define a ring. This, I think, is simply do to school’s emphasis on the real numbers. While I’m not suggesting we throw out the real number system, but it does mean that when people hear of an additive identity, their first thought is “what is that?” and their second is “well, isn’t that just 0?”. Well, it is in fact, not always zero.

I’m going to start with the precise definition first, as I think it will provide better understanding before actually giving the simple version. Thus, the axiom is: There is an element o in R such that a+o=a=o+a for every a in R. Now, you might notice that I didn’t use a zero when writing the formula, but instead used the letter o instead. This is because I want to stress the fact that this isn’t necessarily zero. Even though this is zero in the real numbers, this isn’t necessarily true for all number systems. The simple version is that a number exists which, when added to any other number a, gives us back that other number a. In the real numbers, this is 0, but in other systems , it might not be.

Thinking of an off-the-wall example, let’s say that we are using a number system based off the real numbers, and it goes from 1-6. What is the Additive Identity? Well, the Additive Identity in this case is 6. Why? For an example, let’s look at 1+6. 1+6=7 in our real numbers, but there is no seven. However, what if our number system loops, like going 1, 2, 3, 4, 5, 6, 1, 2, 3, etc. Then, 1+6=1. This holds for all, and thus, 6 is our additive identity for this made up number system. Also, notice that zero wasn’t even a part of the number system in question, so if an additive identity existed, it would have to be something else.

This video discusses not only this axiom, but also some other axioms. Do note that the video only uses the real numbers as an example, so if your not using real numbers, this might change. Abstractly though, this is the same information as above.