# Is exercise 2.22 (iii) in Rotman's Homological Algebra book wrong?

I believe that the statement of exercise 2.22 in Rotman's Introduction to Homological Algebra is wrong. It states that if $$R$$ is an integral domain, $$M$$ is an $$R$$-module, and $$\textrm{Hom}_{R}(M,R/I) = 0$$ for all non-zero ideals $$I$$ of $$R$$, then $$\textrm{Hom}_R(M,R) = 0$$.

If $$R$$ is a field, then the only non-zero ideal is $$(1)$$ and $$\textrm{Hom}_{R}(M,R/(1))$$ is trivially zero, but $$\textrm{Hom}_R(M,R)$$ need not be.

Am I right that the exercise is mistaken? I did not find a correction in the errata.

Edit: Thank you metalspringpro for your answer. The exercise is correct when we exclude the possibility that $$R$$ is a field. To see this one can use the following statement: If the intersection of all non-zero ideals of an integral domain R is non-zero, show that R is a field.

Yes, but it's a rather small mistake; the case you point out is the only one where there is an issue. To fix it, change the claim to

If $$R$$ is an integral domain and not a field, $$M$$ is an $$R$$-module, and $$\operatorname{Hom}_R(M,R/I)=0$$ for all non-zero ideals of $$R$$, then $$\operatorname{Hom}_R(M,R)=0$$.

• Maybe it's just me being stupid, but I don't see how this resolves the issue. The property $\operatorname{Hom}_R(M,R/I)=0$ is always true for $I=(1)$, so what difference would it make to remove $I=(1)$ from consideration? As you wrote, when $R$ is a field, the hypothesis “$\operatorname{Hom}_R(M,R/I)=0$ for all nonzero proper ideals $I$” is vacuously true (since there are no such ideals), but it's still true, so then again a true hypothesis leads to a false conclusion, just as the original true hypothesis “$\operatorname{Hom}_R(M,R/I)=0$ for all nonzero ideals $I$” did. Sep 13, 2021 at 5:26
• @HansLundmark That's a good point, it's now fixed. The point is that you just have to exclude the case where R is a field. I tried to do it in a cute way, but you are correct this doesn't quite work. Sep 13, 2021 at 5:38