An old splitting theorem for (Hausdorff) locally compact abelian (LCA) groups says that any LCA group $L$ is isomorphic to a direct product of $\mathbb{R}^n$ and $L_1$, where $L_1$ contains a compact-open subgroup (this factorization is unique up to isomorphism of $L_1$).

Taking $L = L_1\times\mathbb{R}^n$ as above, we have the following properties:

(i) There is no embedding of $\mathbb{R}^n$, or any closed subgroup of $\mathbb{R}^n$, into $L_1$.

(ii) If $f:\mathbb{R}^n\to L_1$ and $g:L_1\to\mathbb{R}$ are continuous homomorphisms, then $fg$ is trivial.

Does anyone happen to know if the following generalization has been studied, or if there are some other known examples:

An additive category $\mathcal{C}$ such that every object $A$ can be expressed as a product $A = B\times C$ where:

(i') There is no embedding of any sub-object of $C$ into $B$.

(ii') If $f:B\to C$ and $g:C\to B$ are morphisms, then $fg$ is trivial (maps onto the zero-object).

EDIT: Reworded in terms of additive categories, not abelian.