Axiom 10: Multiplicative Identity

If you watched the podcast on the history of Ring Theory, then you should recognize this axiom as one of the main differences between Fraenkel and Noether. Now, the reason that this isn’t part of the base axioms is simply that base 8 axioms are the ones that are the minimum required for a ring to resemble the integers. This axiom is part of the “additional” axioms, and this one in particular identifies a ring that we can call “A Ring with Identity”.

A Ring with identity has the following property: a Multiplicative Identity. What is a multiplicative identity? A multiplicative identity can be defined in the following manner: a ring with identity is a ring R that contains an element “1” satisfying a”1” = a = “1”a for all a in R. Now, why did I put “1” in quotation marks? After seeing this, I’m sure many of you were thinking “doesn’t 1 always give you the same result back when you multiply?” This statement is true, but only when we are considering the real numbers, which is probably the number system that you were referring to when you thought that. You see, I put “1” in quotation marks because in many number systems, 1 isn’t actually the multiplicative identity. In fact, a multiplicative identity is any number that when you multiply another number by it, you get back that number.

So, in what systems would 1 not be the multiplicative identity? I’d like you to consider the number system 0, 2, 4, 6, 8. Here, in this number system, 1 isn’t even an element, so it can’t even be considered as a possible multiplicative identity since it doesn’t even exist in the number system. In fact, in this number system, multiplying two numbers would take it outside of the number system, so let’s define it further. Let’s say that, when multiplying, the number gotten is the number in the ones place. For example, 6x8=48, so in this system, 6x8=8. Does this clarification give us a multiplicative identity? If you got 6, you’re correct! 6 times any of the numbers result in that number, which is exactly the definition of a multiplicative identity.


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.