If you can define division "trivially" to make a system into a Field.

Question

Thinking of multiplication outside of numbers is pretty straightforward, but division is tricky. That led me to the notion of Division Rings, and got me wondering if you could instead define division "trivially" to make it into a field.

What I mean by that is, division is undefined for certain numbers, mainly divide by zero:

$$\frac{x}{0}$$

But what if you just said that divide by anything is undefined. Or to take it only half way, say we have 2 values in our system $a$ and $b$. We can only divide by $a$. Wondering if that is enough to say "we have division" to allow the definition of a field to be applied. Then say we have 3 values, and we can only divide by 1 of them, so 33%, then 25%, then ... 0.000001% of the values we can divide by. At some point we could say that we can't divide by anything, yet division is still defined (to make it into a field). Division by the "empty" value, let's say.

Wondering if that works or how I should think about that. Also wondering, if it is possible to define division like this, what you lose when thinking about the system in terms of a field.

Answer 1

To be a field, you have to be able to divide by everything nonzero. If you can't do that, then you don't have a field.

However, there are more general constructions involving fractions. It sounds like you are beginning to conceive of the localization of a ring: we select a a multiplicative subset of elements that are allowed to be denominators, and construct the algebraic structure consisting of fractions with arbitrary numerator and just those denominators.

Answer 2

If you want to construct a field out of a ring you have to define fractions and add them to your ring.

Now you have to define fractions carefully so they are well defined.

For example in the ring of integers, you define an equivalence relation on $$Z\times (Z-\{0\})$$ namely $$ (a,b) R (c,d) \iff ad=bc $$ and define $a/b$ to be the equivalence class of $(a,b).$

This way you have $$ 2/3 =4/6=12/18=.....$$ and everything starts making sense because the field of rational numbers is born out of the ring of integers.