If R is an infinite direct product of fields, then R is an injective R-module...But I need an example of a quotient ring R, R/S, that is not injective ?? I feel that "if we take S as an infinite direct sum of fields then R/S may not be injective ??? But I couldn't show it...**Please help me to find R/S which is not injective**.



a quotient of an injective module which is not injective
(I just stumbled upon this question from a link that David White listed at a [completely different question](https://mathoverflow.net/questions/75418/commutative-algebra-final-project).)


I can give a simple argument that shows some noninjective quotient of $R$ does indeed exist. Perhaps folks already know this, but nobody seems to have said it.


A famous result of Osofsky states that a ring is semisimple if and only all of its cyclic modules are injective:


[B. Osofsky, *Rings all of whose finitely generated modules are injective*, Pacific J. Math., 1964](https://projecteuclid.org/journals/pacific-journal-of-mathematics/volume-14/issue-2/Rings-all-of-whose-finitely-generated-modules-are-injective/pjm/1103034192.full)


(See the Theorem, p. 649.) Certainly this ring $R$ is not semisimple (as it's not even noetherian). So *some* quotient module $R/I$ must be noninjective.


I certainly think that the choice of $I$ above is as good a guess as any. If I have some time to think about whether it is indeed noninjective, I'll come back and edit this.


