How to name these "ideals"? 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 $I \cdot M = 0$.
 A: I have just named then idals in the paper. They can be also called co-well-pointed objects, which is a bit too long, though.
A: I'll suggest a slight generalization of your point of view that might be of interest (at least in order to formulate a name): consider the case $\mathcal{C} = \mathsf{Mod}(\mathbb{Z}) = \mathsf{Ab}$ is the category of abelian groups. 
In that case, as you pointed out, $\mathbf{1} = \mathbb{Z}$ is a ring (as any monoid object in $\mathsf{Ab}$ is a unital, non-commutative ring), and all objects $g : \mathsf{Ab}$ are $\mathbf{1}$-bimodules: they have structural maps $\lambda : \mathbf{1} \otimes g \to g$ and $\rho : g \otimes \mathbf{1} \to g$ (here, $\lambda$ and $\rho$ stand for left and right unitor, but it is harmless to consider them as standing for left and right action).
In this context, what you described defines an ideal of $\mathbb{Z}$, namely the image (or cokernel) of $e : E \to \mathbf{1}$. The problem is that this correspondance is not 1-1, even if we consider isomorphism classes of morphisms: this is due to the fact that the kernel of $e$ is the pushout out of the span $* \leftarrow E \xrightarrow{e} \mathbf{1}$ (where $*$ is the terminal object) and this pushout is not 1-1 with $e : E \to \mathbf{1}$.
In this spirit we could call these things "generalized ideals", where "generalized" only means that they correspond to ideals in a non-unique way. A more "sensible" name might come to mind when you ask how this correspondence can fail to be 1-1: since all kernels are submodules and $Im(e) \cong E/Ker(e)$ maybe "sup-ideals" could be more intuitive, meaning that some quotient of them is (isomorphic to) an ideal.
There is a general argument by which one can reconstruct your statement about arbitrary (unital, commutative) rings from the above fact about $\mathbb{Z}$, that can be summarized as follows (motivated by the fact that monoids in $\mathsf{Ab}$ are unital rings, and commutative monoids are commutative unital rings):
If we fix another monoid $(M, \epsilon, \mu)$ in $\mathcal{C}$ any monoidal category we can consider the category of bimodules over it $\mathsf{Bim}_{M}$: if $M$ is commutative and $\mathcal{C}$ is symmetric and cocomplete then so is $\mathsf{Bim}_{M}$ (see here for a reference). Moreover, cokernels always exist (in $\mathcal{C}$ and hence in $\mathsf{Bim}_M$) if $C$ is cocomplete (as they are pushouts out of the span sketched above, hence colimits), so it should be possible to generalize the idea for $\mathbb{Z} = \mathbf{1} : \mathsf{Ab}$ to arbitrary monoid objects $R$ by replacing $\mathsf{Ab}$ with $\mathsf{Bim}_R$ and to arbitrary symmetric, cocomplete monoidal categories $\mathcal{C}$.
