When working with injective modules, one bad thing is that they do not necessarily behave well with respect to localisation. Consider a commutative ring $R$ and have a look at the following properties:
(1) If $I$ is an injective $R$-module, then $I_{\mathfrak{p}}$ is an injective $R_{\mathfrak{p}}$-module for every prime ideal $\mathfrak{p}\subseteq R$.
(2) If $I$ is an $R$-module such that $I_{\mathfrak{p}}$ is an injective $R_{\mathfrak{p}}$-module for every prime ideal $\mathfrak{p}\subseteq R$, then $I$ is injective.
For an arbitrary ring, neither of these properties are necessarily true.
I know of two classes of rings for which both (1) and (2) are true; namely, noetherian rings, and $h$-local orders in semisimple rings (e.g. $h$-local domains).
In addition, I know two further classes of rings that satisfy (1); namely, hereditary rings, and polynomial algebras in countably many indeterminates over noetherian rings. The latter rings do not necessarily satisfy (2), and of the former I do not know whether or not they do.
This leads me to the following question:
Which other classes of rings satisfying (1) and (2), or at least one of these properties?
Some references: E.C.Dade, Localization of injective modules, J. Algebra 69 (1981), 416-425; F. Couchot, Localization of injective modules over valuation rings, Proc. Amer. Math. Soc. 134 (2005), 1013-1017; C. Naudé, G. Naudé, On localization of injective modules, J. Algebra 103 (1986), 108-115.
