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.

  • $\begingroup$ Pardon me for my ignorance but I'm wondering whether you need $R$ to be Noetherian for (1) and (2) to be true? $\endgroup$ – Daniel Pomerleano Aug 29 '13 at 12:07
  • $\begingroup$ I guess the answer must be no based upon the rest of the proof, but what is a proof/reference? $\endgroup$ – Daniel Pomerleano Aug 29 '13 at 12:08
  • $\begingroup$ Dear @Daniel, I added some references. $\endgroup$ – Fred Rohrer Aug 30 '13 at 4:19
  • $\begingroup$ Thanks! I had in mind the standard proof that localization at any multiplicative set preserves injectives for Noetherian rings, and remembered that the Noetherian hypothesis is very important. However, in Dade he is able to prove this for localizing at prime ideals. $\endgroup$ – Daniel Pomerleano Aug 30 '13 at 12:54
  • $\begingroup$ Dear @Daniel, do you claim that Dade shows that localisation at primes preserves injectivity over arbitrary rings? If so, where? $\endgroup$ – Fred Rohrer Aug 30 '13 at 20:07

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