Endomorphism ring as a set of matrices

Question

Let $A=\mathbb Z[\sqrt{-5}]$, and let $I=(2,1+\sqrt{-5})$ (which is known to be a non-principal ideal of $A$ with $I^2=2A$). If we put $P=A \oplus I$, my question is:

Why the endomorphism ring of $P$ is the subring of matrices $\begin{pmatrix}A & I^{-1} \\I & A \\\end{pmatrix} \subseteq \mathbb M_2(K)$, where $K$ is the quotient field of $A$, and $I^{-1}=\{a\in K: aI \subseteq A\}$?

Thanks in advance!

Answer 1

We have $\operatorname{Hom}_A(A,A)\simeq A$, $\operatorname{Hom}_A(A,I)\simeq I$, $\operatorname{Hom}_A(I,A)\simeq I^{-1}$, and $\operatorname{Hom}_A(I,I)\simeq A$. Is this enough for you to conclude?