For a large part of its history, commutative algebra was viewed as studying arithmetic more generally (in objects called rings), and solving equations in these new contexts (in objects called modules). Grothendieck's relative perspective changed the way mathematicians thought about commutative algebra in the late 20th century. Rings could be viewed as functions on a space of its prime ideals. In this way, many algebraic notions in rings can be interpreted purely topologically. This talk will hopefully provide you with a new perspective on rings, starkly different from the one shown in GRM.
In this talk we will discuss localisations of rings and the wonderful theorems they produce. After building some familiarity with these concepts, we will briefly discuss a geometric interpretation through the spectrum of prime ideals in a ring. The entire talk will be done constructively; without use of law of the excluded middle or Zorn's lemma.

Prior knowledge on rings is required.

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