The motivation behind axiomatisation

Axiomatisation in the context of rings

I am in the middle of an elementary pure mathematics unit and have just started looking at the concept of rings. In lectures, we have divided up rings into several categories defined axiomatically:

• Commutative rings - multiplication is commutative but there does not exist a multiplicative inverse for every non-zero element.
• Division rings - a multiplicative inverse exists for all non-zero elements, but multiplication is not commutative.
• Fields - multiplication is both commutative and invertible.

I can understand the motivation to separate and 'axiomatise' these structures so that their properties are inherited by 'sub'-structures which are consistent with their axioms (i.e. quaternions form a division ring and so they must have certain properties), but I am still confused about the purpose of this practise.

So here are the questions:

1. Why do mathematicians define these sorts of structures?
2. How do they aid pure mathematicians in their research?
3. What are some examples of non-obvious properties that can be inferred from these or other axioms?

I have some experience programming so if analogies can be drawn between this and object oriented programming (i.e. subclassing perhaps) then that could help. Also please keep in mind that I have only just started learning abstract algebra so I will not be familiar with many structures or advanced concepts. Thanks.

• Ah right, I keep mixing that up. Thanks. Apr 8, 2014 at 15:11

You might be familiar with the programming rule of abstraction that the third time you write a piece of code to do something, you should refactor your program to move that functionality out into its own class or function.

Commutative rings, division rings, and fields have come up way more than three times in mathematics. :)

Encapsulation is just as useful in mathematics as in programming to hide away extraneous information, thus making it easier to work with something. The best example of this sort of thing I know came up in quantum physics (which I will describe somewhat glibly): when the field began, all of the math was deep in the theory of partial differential equations and differential operators, which are quite complicated and rather laborious to work with.

Then Dirac came along and said "this is all just linear algebra", introduced Hilbert spaces and bra-ket notation, and poof quantum physics quickly became much, much easier to work with and reason about.

My history is somewhat weak, but I believe commutative algebra really originated with the development of algebraic number theory: when people started studying number rings and number fields in the attacks on Fermat's Last Theorem, they quickly realized that there was a lot of interesting structure there... as well as lack of structure (e.g. many number rings don't allow unique factorization into elements).

As new structure was discovered, new ways rings could be pathological, and new sorts of arguments for reasoning about rings, definitions were given that supported these sorts of structures and reasoning, or separated the rings we are actually interested in from the badly behaved ones.

e.g. the notion of a Noetherian ring came about when Emmy Noether realized the power of making arguments based on the ascending chain condition: an infinite chain of ideals $I_1 \subseteq I_2 \subseteq \cdots$ is eventually constant. Rings people are interested in often have this property, but general rings do not: the structure "Noetherian ring" is thus defined, so that people have words to say that they are working with rings that allow this sort of argument.

• You've explained well the abstraction aspect of axiomatisation and how historically it makes algebraic manipulation less arduous, but I'm also looking for a simple, concrete example (that I am capable of understanding at this stage!) of how a significant theorem or other result is derived from axioms and applied across a range of sub-structures as a result of 'axiomatic encapsulation'. Preferably a result that was discovered a reasonable period of time after the invention of the 'axiomatic superstructure'. I ask for this because I want a tangible sense of how these definitions help. Apr 8, 2014 at 15:26