It is known that if a commutative Noetherian ring $R$ is hereditary then for any maximal ideal $M$ the localization $R\_M$ is also hereditary. Is the Noetherian assumption necessary? 



Localizations of hereditary rings
You do not need to suppose your ring to be noetherian. In fact, the following stronger statement is true:



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> If $R$ is a hereditary commutative ring and $S\subseteq R$ is a subset, then the ring $S^{-1}R$ is hereditary.
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In order to prove this we use the following two facts:


1) A commutative ring $R$ is hereditary if and only if any epimorphism of $R$-modules with injective source has injective target (i.e., quotients of injectives are injective). (See e.g. T.Y. Lam, *Lectures on modules and rings,* Theorem 3.22.)


2) If $R$ is a commutative ring, $S\subseteq R$ is a subset an $M$ is an $S^{-1}R$-module, then $M$ is injective if and only if it is so considered as an $R$-module by means of scalar restriction. (See e.g. M.P. Brodmann, R.Y. Sharp, *Local cohomology,* Lemma 10.1.12 (where the noetherian hypothesis is not used), or E.C. Dade, *Localization of injective modules,* J. Algebra 69 (1981), 416--425.)


Now, suppose that $R$ is a hereditary commutative ring and consider a subset $S\subseteq R$. Let $M\rightarrow N$ be an epimorphism of $S^{-1}R$-modules with injective source. By means of scalar restriction to $R$ we can consider this as an epimorphism $M\rightarrow N$ of $R$-modules with injective source by 2). Hence, its target is injective by 1), and therefore the $S^{-1}R$-module $N$ is injective by 2). It follows that $S^{-1}R$ is hereditary.


