For $R$-module category, we have the notion of split exact sequence which is given by the following.
For a fixed ring $R$, let $M,M^\prime,M^{\prime\prime}$ be modules over $R$.
We call an exact sequence $0\rightarrow M^{\prime\prime}\stackrel{j}{\rightarrow} M\stackrel{\pi}{\rightarrow} M^\prime\rightarrow 0$ split if there exists a map $s:M^\prime\rightarrow M$ such that $\pi\circ s=id_{M^\prime}$.
In this case, we know that there exists a map $t:M\rightarrow M^{\prime\prime}$ such taht $t\circ j =id_{M^{\prime\prime}}$ and $M=M^{\prime\prime}\oplus M^\prime$.
My question is, if we consider the notion for rng category instead of module category over a fixed ring, do we have the same result?
Notice that the notion of split exact sequence is not well defined in ring category but we can consider this notion in rng category. Here is the reason why:Some exact sequence of ideals and quotients
To be more specific, if $0\rightarrow J \stackrel{j}{\rightarrow} A\stackrel{\pi}{\rightarrow} B\rightarrow 0$ is a split exact sequence in rng category, which means that there exists a map $s:B\rightarrow A$ such that $\pi\circ s=id_{B}$, do we know that $A=J\oplus B$?
I guess the answer should be NO.
For example, if $J$ is a rng without unit and we let $J^{+}$ be the utilization of $J$, then we have an exact sequence $0\rightarrow J \stackrel{j}{\rightarrow} J^+\stackrel{\pi}{\rightarrow} \mathbb{Z}\rightarrow 0$, where $j:J\rightarrow J^+$, $j(a)=a+0$, $\pi:J^+\rightarrow \mathbb{Z}$, $\pi(a+\lambda)=\lambda$. We can easily verify that $s:\mathbb{Z}\rightarrow J^+$, $s(\lambda)=(0+\lambda)$ satisfies the condition $\pi\circ s=id_{\mathbb{Z}}$.
However, we know that $J^+\ncong J\oplus \mathbb{Z}$, since $J^+$ is a unit ring and $J\oplus \mathbb{Z}$ doesn't have a unit. We can also verify that there won't be any ring homomorphism $t:J^+\rightarrow J$ such that $t\circ j=id_{J}$.
My question is: Is this proof right? If not so, where did I go wrong?
If the proof is right, then the notion of split exact sequence in rng category seems not so good, since the three conditions listed before are not equivalent. (It is well known that the three conditions are equivalent in $R$-module category). In this case, I wonder in which category we can define the notion of split exact sequence nicely?
Any help will be truly grateful!