An $R$-linear map out of $R$ is determined uniquely by the image of $1$.Suggest a slogan
Let \(R\) be a ring with \(1\) and \(A\) a left \(R\)-module. The map \(\Hom_R(R, A)\to A\), \(f\mapsto f(1)\), is an isomorphism of abelian groups, with inverse sending \(a\in A\) to the homomorphism \(r\mapsto ra\); if \(R\) is commutative, it is an isomorphism of \(R\)-modules.
What is $\mathop{\mathrm{Hom}}_R(R, A)$ for a left $R$-module $A$?