The Stacks project

Exercise 111.64.7 (Injectives). Let $(A, \mathfrak m, \kappa )$ be an Artinian local ring. Assume $A$ is injective as an $A$-module. Show that $\mathop{\mathrm{Hom}}\nolimits _ A(\kappa , A)$ has dimension $1$ has a $\kappa$-vector space.