Axiom 7: Associative Multiplication

Just like with the Associative Addition Axiom, the Associative Multiplication Axiom is a property that we have all been taught before, even when we didn’t have a name to give to the property. As such, it’s hard to imagine something this basic not working. It intuitively makes sense for us to use this property, since it functions with our everyday real numbers.

Now, before I get ahead of myself, what is the Associative Multiplication Axiom? In essence, this axiom says that the following two expressions are equivalent: a(bc)=(ab)c where a, b, and c are in the ring. Now, in English, what does this tell us? This says that when we take any three numbers in our numbers system we are using (the real numbers, integers, or anything else), we can multiply the numbers in any order. This makes intuitive sense, I believe, to all of us, just as it did when we were looking at the addition equivalent.

Thankfully, when we are trying to prove whether or not a number system is a ring, this property is relatively easy to prove, especially when trying to compare to the other axioms that are on this list. Similar to the additive associative axiom, we usually can tell pretty quickly if this property will hold. Allow me to use the real numbers as an easy example. Does multiplying real numbers together actually matter on the order? We all know that order doesn’t matter for the real numbers, and thus it is shown that this is true.


This video gives another example and voice as an explanation for this axiom.