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At the very beginning of Ravi Vakil's amazingly famously amazing and famous notes on algebraic geometry, he remarks that some familiarity with localization and prime ideals is useful. I don't know anything about commutative algebra, so I was hoping for a categorical interpretation of both these concepts, or at least a nice conceptual summary from a structural viewpoint. Sources and references kindly accepted (as low level as possible please).

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The localization of a ring $A$ at a multiplicative submonoid $S$ is the initial ring under $A$ in which $S$ is sent to units. (i.e. there's a localization map $A\to S^{-1}A$ and if $S$ goes to units in $f:A\to B$ then $f$ factors uniquely through $A\to S^{-1}A$.) Diagrammatically, a unit is just an element whose action by multiplication is an isomorphism.

A prime ideal $\mathfrak{p}$ is one for which the quotient $A/\mathfrak{p}$ is an integral domain, that is, a ring in which all of the endomorphisms given by multiplication by an element are monomorphisms. In this interpretation it's not immediately obvious that the complement of a prime ideal is a multiplicative submonoid, but it is, and this is the most important kind of submonoid at which one localizes.

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Any ring can be viewed as an additive category with one object and the endomorphisms of that object given by the elements of the ring. Then Gabriel-Zisman localization applied to this special case recovers the usual localization of a ring at a multiplicative subset of elements. This makes sense even when the ring is not commutative, and then one recovers Ore localization of an associative ring. As a reference for Gabriel-Zisman localization, I would recommend the classic book with the same authors, called "Calculus of fractions and homotopy theory".

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  • $\begingroup$ Is the additive structure of the categorification of a ring what sets it apart from the categorification of its underlying monoid? $\endgroup$
    – user153312
    Dec 17, 2014 at 20:03

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