Comment by aleph_minus_one

If you just want to do some stupid computations: sure.

But this is not what mathematics is centrally about. The central point is the kind of thinking about the respective topics (and understanding it) which these more abstract definitions encode.

Understanding the topic just enough to do some elementary computations does not give you the kind of thinking that is often near a transcendental experience.

Just to give one example: the reason why the localization of a commutative ring (a generalization of the field of fractions) is introduced is that many properties of ring hold if and only if they hold for all of its local rings; see for example [1]. This means to understand some property of a commutative ring R, we "just" have to understand its (simpler) local rings.

This is an example why one wants to study such ideas; on the other hand, I can imagine sooo many more exciting things to do with my life than dividing numbers by each others to form fractions. :-)

[1] https://en.wikipedia.org/w/index.php?title=Localization_(com...