The minimal primes of localizations of rings

Abstract

The set of minimal primes of a ring is a very important set as far the spectrum of a ring is concerned as every prime contains a minimal prime. So, knowing the minimal primes is the first (important and difficult) step in describing the spectrum. In the algebraic geometry, the minimal primes of the algebra of regular functions on an algebraic variety determine/correspond to the irreducible components of the variety. The aim of the paper is to obtain several descriptions of the set of minimal prime ideals of localizations of rings under several natural assumptions. In particular, the following case are considered: a localization of a semiprime ring with finite set of minimal primes; a localization of a prime rich ring where the localization respects the ideal structure of primes and primeness of certain minimal primes; a localization of a ring at a left denominator set generated by normal elements, and others. As an application, for a semiprime ring with finitely many minimal primes, a description of the minimal primes of it's largest left/right quotient ring is obtained. For a semiprime ring $R$ with finitely many minimal primes $\min (R)$, criteria are given for the map $$ρ_{R,\min}: \min (R)\to \min (Z(R)), \;\; \mathfrak{p} \mapsto \mathfrak{p}\cap Z(R)$$ being a well-defined map or surjective where $Z(R)$ is the centre of $R$.