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:
- Why do mathematicians define these sorts of structures?
- How do they aid pure mathematicians in their research?
- 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.