What is...localization?

Goal.
Explaining basic concepts of (a classical course in) algebra in an intuitive way.

This time.
What is...localization? Or: Why fractions matter.

Disclaimer.
Nobody is perfect, and I might have said something silly. If there is any doubt, then please check the references.

Slides.

Material used.

Fractions.

Pictures used.

#algebra
#galoistheory
#mathematics

[MncarK]

Great video. But when I saw the video at the end, I had doubts about the set S, shouldn't S be containing s(a) ≠ 0 (otherwise [1] can not be contained in S ) ? Sorry to bother.

[jimnewton]

in all the examples I've seen, we start with some ring R, and a multiplicative monoid S. Is there some reason that S needs to be a subset of R? In the examples it always is. But does it need to be?

[moritzkranitzer]

I really enjoyed this video! Thanks you very much :)

[jimnewton]

In all summaries of localization which I've seen thus far, we take some ring R and a set S which is a "multiplicatively closed set which contains 1". Why don't people just say ring R and monoid S? Is a "multiplicatively closed set containing 1" somehow subtly different than a multiplicative monoid? Why invent a new word to describe a known, well understood concept? Or is the distinction really important?

Is it that you want to allow multiplication to be non-associative on S, or you want to support cases where S is NOT a subset of R?