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

If $R$ is a hereditary commutative ring and $S\subseteq R$ is a subset, then the ring $S^{-1}R$ is hereditary.

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.
