*Background.* Let $\mathcal{C}$ be a symmetric monoidal category with unit $\mathbf{1}$. A subobject of $\mathbf{1}$ is just a monomorphism $I \to \mathbf{1}$. We may also call this an ideal of $\mathbf{1}$, because if $\mathcal{C}=\mathsf{Mod}(R)$, where $R$ is a commutative ring, we obtain the usual notion of an ideal of $R$.

In my research a certain generalization of this concept appears, namely a morphism $e : I \to \mathbf{1}$ such that $e \otimes I = I \otimes e : I \otimes I \to I$.

What short name for this kind of object do you suggest? It's like an ideal, but something is missing, namely $e$ is not a monomorphism. I'm not really content with "idal". Or do these objects already have a name and do they appear in the literature?

The product of two "idals" $e : I \to \mathbf{1}$ and $f : J \to \mathbf{1}$ is just $e \otimes f : I \otimes J \to \mathbf{1} \otimes \mathbf{1} \cong \mathbf{1}$. One might define the sum as $(e,f) : I \oplus J \to \mathbf{1}$, but this is no "idal". If $\mathcal{C}=\mathsf{Mod}(R)$, then an "idal" is an $R$-module $I$ with an $R$-linear map $e : I \to R$ such that $e(x) \cdot y=e(y) \cdot x$ holds for all $x,y \in I$.

Edit: I have found a classification in the case of $R$-modules over a Dedekind domain $R$. Here every idal is isomorphic to $I \oplus M \twoheadrightarrow I \hookrightarrow R$ for some ideal $I \subseteq R$ and some $R$-module $M$ such that $I \cdot M = 0$.