Let $(A, P)$ be a zero-dimensional local ring with maximal ideal $P$. Suppose that for some local ring $(B, P_B)$, $A$ is $B$-algebra that is finitely generated as a $B$-module and the maximal ideal of $B$ maps into that of $A$. Let $E$ is the injective hull of the residue class field of $B$. The 'injective hull' is defined as follows ( Eisenbud's Commutative Algebra book proposition-Definition A.3.10 ):
Definition 1. Let $R$ be a ring and let $M \subset E$ be $R$-modules. We say that $M$ is an essential submodule of $E$, or that $E$ is an essential extension of $M$ if every nonzero submodule of $E$ intersects $M$ nontrivially.
Proposition-Definition A3.10. Let $R$ be a ring. There is, up to isomorphism, a unique essential extension $E$ of $M$ that is an injective $R$-module ; this $E$ is called the injective envelope ( or injective hull ) of $M$, writtten $E(M)$.
Consider $\bar{E}:=\operatorname{Hom}_B(A,E)$, endowed with the $A$-module structure as follows : "For $a\in A$ and $\varphi \in \bar{E}$, $a\varphi ( a') := \varphi(aa')$ for all $a' \in A$." ( This is really structure of $A$-module ? ).
Let $N\subset \bar{E}$ be a nonzero $A$-submodule. So there is a nonzero $0 \neq \varphi : A\to E \in N$. Note that since $A$ is zero-dimensional, its maximal ideal $P$ is nilpotent ideal ( C.f. Theroem 2.12 in the Eisenbud's book ). So $P^{k} =0$ for some natural number $k\in \mathbb{N}$.
Q. Then my question is, does there exist $0 \neq \varphi' \in N $ and $a \in A$ such that $a \varphi' \neq 0 $ and $P(a\varphi')=0$ ( annihilated by $P$ ) ? Or more strongly, for any nonzero $\varphi \in \bar{E} := \operatorname{Hom}_B(A,E)$, does there exist $0\neq a\in A$ such that $a\varphi \neq 0$ and $P(a\varphi)=0$?
Let $k \in \mathbb{N}$ be the smallest number such that $P^k= 0$. Choose $0 \neq a \in P^{k-1}$. Then maybe $P(a\varphi)=0$. An issue that makes me stuck is possibility of $a\varphi =0$. I cant' find such suitable $a\in A$ at all until now. I am struggling with this issue with hours.
This question originates from the final paragraph of proof of the Eisenbud's Commutative Algebra book, Proposition 21.4 : (C.f. In proof of the Proposition 21.4 in the Eisenbud's Commutative Algebra book ( First question, Canonical module of a local zero-dimensional ring ). ). As an image,
Why the underlined statement is true?
Can anyone help?
