Let $k$ be a commutative ring.

Let $C$ be a filtered $k$-coalgebra. This means a $k$-coalgebra equipped with an increasing $k$-module filtration $C^0 \subseteq C^1 \subseteq C^2 \subseteq ...$ satisfying $\bigcup\limits_{n\geq 0}C^n = C$ and $\Delta\left(C^n\right) \subseteq \sum\limits_{k=0}^n C^k \otimes C^{n-k}$ for all nonnegative integers $n$, where $C^k \otimes C^{n-k}$ really means the image of $C^k \otimes C^{n-k}$ inside $C \otimes C$ by abuse of notation (even if the map $C^k \otimes C^{n-k} \to C \otimes C$ fails to be injective).

Assume further that $C$ is connected, i. e., the counit $\varepsilon$ restricted to $C^0$ is an isomorphism to $k$. This allows us to define an element $1_C \in C^0$ (often just called $1$ when no confusion can arise) as the preimage of $1 \in k$ under this isomorphism (even if $C$ has no algebra structure given).

An element $x\in C$ is said to be primitive if $\Delta x = x \otimes 1_C + 1_C \otimes x$. The $k$-submodule of $C$ formed by all primitive elements is called $\mathrm{Prim} C$. Every primitive element $x\in C$ satisfies $\varepsilon x = 0$ (this is easy to check).

Conjecture 1: If $D$ is a further $k$-coalgebra, and $f : C \to D$ is a homomorphism of $k$-coalgebras such that $f\mid_{\mathrm{Prim}C}$ is injective, then $f$ is injective.

Conjecture 2: If $I$ is a coideal of $C$ such that $I \cap \mathrm{Prim}C = 0$, then $I = 0$. (Here, a coideal of $C$ means a $k$-submodule $J$ of $C$ such that $\Delta J \subseteq J \otimes C + C \otimes J$, where again an abuse of notation is made.)

Background: Conjecture 2 is equivalent to the special case of Conjecture 1 when $f$ is assumed surjective. Both conjectures are known to be true if $k$ is a field, and the easy proof (by induction) generalizes to the case when "everything in sight is flat" (I don't want to go into the details of what this means, but I can give the proof if needed). Moreover, Conjecture 2 is also true when $C$ is a graded coalgebra and $I$ is homogeneous (more or less because direct sums are just as good as flatness when it comes to preserving injectivity under tensor products). I am unable to come up with a counterexample in any other case, but this is because I have no real idea how to construct non-flat counterexamples to anything (a good reference for non-flat perversion would be highly appreciated). It seems to me that the conjectures should be true, if only because they are highly useful.