An R-linear map R/I to R I've found the following statement in the K. Conrad's note, Example 4.4 (http://www.math.uconn.edu/~kconrad/blurbs/linmultialg/noetherianmod.pdf):
Let $I$ be an ideal in a ring $R$, then $Hom_R(R/I, R) \cong \{r\in R: Ir=0 \}$. 
Why this is true?  

Say you want to construct a map $\varphi:R/I\to R$ which is $R$ linear. As suggested in the comments, you just need to choose $\varphi(1)$. The rest is determined automatically by $R$ linearity. By definition of the quotient, the element $1\in R/I$ is annihilated by $I$. Hence, $\varphi(1)$ also has to be annihilated by $I$.
In general, if $M,N$, are $R$ modules, $\varphi:M\to N$ is $R$ linear, and $m\in M$ is annihilated by the ideal $I$, then so is $\varphi(m)$. Indeed, for $i\in I$ we have $$i\varphi(m)=\varphi(im)=\varphi(0)=0.$$
