Given a ring $A$ and an $A$-module $M$, there is a unique (up to isomorphism) injective $A$-module $E_A(M) $containing $M$, with the property that every injective $A$-module containing $M$ also contains $E_A(M)$. $E_A(M)$ is called the injective hull or injective envelope of $MS.
Example 2.32 from Richard Stanley's Combinatorics and Commutative Algebra, page 16: Let $A = k[x_1,..., x_n]$, where $k$ is a field. Regard $k$ as an $A$-module via the isomorphism $k \cong A/(x_1 A + ... + x_n A)$, Then $E_A(k) \cong k[x_1^{-1},..., x_n^{-1}]$.
Why is $k[x_1^{-1},..., x_n^{-1}]$ the injective hull of $k$ considered as $A$-module? Is there a conceptional way (eg by using some structure results) to show it? Why is it even injective?