Recall that an associative algebra $A$ is called *idempotented* provided that is the filtered union of subalgebras $eAe$ for $e \in A$ idempotent. I think sometimes people say that $A$ has approximate identity.

It is well-known that for any idempotent $e \in A$, the functor $M \mapsto eM$ induces a bijection from simple $A$-modules $M$ satisfying $eM \neq 0$ to simple $eAe$-modules. My question is whether this bijection can be deduced from some equivalence of categories. Here's my guess: the aforementioned functor restricts to an equivalence from $A$-modules $M$ satisfying $AeM = M$ to $eAe$-modules. Note that this functor has a left adjoint, namely $N \mapsto Ae \otimes_{eAe} N$, which must be the inverse if my guess is true.

Here's the tricky point: if $M$ is a $A$-module satisfying $AeM = M$, we have to show that $Ae \otimes_{eAe} eM \to M$ is injective. This is not at all clear to me.

Is my guess correct? If not, can it be modified into a true statement? I really only care about the case where $A$ is the Hecke algebra of a $p$-adic group and $e$ is the indicator function of a compact open subgroup, so if there is a counterexample of this kind I'd like to see it.

**Edit:** As Benjamin Steinberg's answer shows, this is *false* in the stated generality. So let me reiterate that I'm interested in Hecke algebras of $p$-adic groups, and ask more directly: is my guess correct in this setting?