Localizations of essential extensions

Abstract

In an earlier paper [4] we considered the question of whether an injective module E over a noncommutative ring R remains injective after localization with respect to a denominator set X in R. A related question is whether, given an essential extension N of an R-module M, the localization N[X–1] must be an essential extension of M[X–1]. In [1] it is shown that if R is left noetherian and X is central in R, then localization at X preserves both injectivity and essential extensions of left R-modules and, hence, preserves injective hulls and minimal injective resolutions.