Is there an easy way to remember the ring axioms?

Question

I'm working on a problem that asks me to prove a set is a ring. I have to look up the axioms to prove it...

I was curious, is there an easy way to remember the ring axioms, so that you can avoid having to look them up?

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The best way I have so far, is simply remembering that there are 4 axioms for addition, and 4 for multiplication.

Answer 1

A ring is a fusion of two very basic structures, namely an abelian group (4 axioms) and a monoid (2 axioms), compatible via distributive laws (2 axioms).

"I'm working on a problem that asks me to prove a set is a ring." - This doesn't mean that you have to verify all ring axioms. In fact, there are many "basic" rings and constructions with rings and often the task is only to realize that something is a subring of such a basic ring. In order to check for a subring, we only have to check:

• Is $0$ contained? Is $1$ contained?
• Is the subset closed under $+$ and $-$?
• Is the subset closed under $*$?

All the other ring axioms are inherited automatically from the ring "above". For example, for any space $D$, the set $\{f : D \to \mathbb{R} : f \text{ continuous}\}$ is a subring of the "basic" ring of all functions $D \to \mathbb{R}$ (with pointwise operations) because of calculus facts about continuous functions: Constant functions are continuous, and continuous functions are closed under $+,-,*$.

Unfortunately, some exercises want you to verify the ring axioms with super-artificially defined rings. For example, the set $R$ of pairs $(a,b)$ of integers $a,b \in \mathbb{Z}$ with addition $(a,b)+(c,d) = (a+c,b+d)$, but multiplication $(a,b) *' (c,d) = (ac+2bd,ad+bc)$. You can spend pages with computations, or you learn the general construction of generated subrings and realize that the above ring is just $\mathbb{Z}[\sqrt{2}]$ (the subring of $\mathbb{R}$ generated by $\sqrt{2}$) in disguise, where $(a,b)$ encodes $a+ b \sqrt{2}$. Since $\mathbb{Z}[\sqrt{2}]$ is a ring for trivial reasons, the same is true for $R$ without any necessary computations.

Answer 2

It may possibly be easier to remember what axioms are missing with multiplication.

As in, multiplication has an identity element, and is associative, but does not have to be commutative, and doesn't need an inverse, whereas addition has all of these things.

That leaves left and right distributivity. And the fact that left and right distributivity are called out separately may help you remember that multiplication doesn't have to be commutative.