I'm not as familiar with constructive proofs, so this "answer" involves a few questions.

Consider the ring $R=\mathbb{M}_2(\mathbb{F}_2)$, and the element $r=\begin{pmatrix} 0 & 1\\ 0 & 0\end{pmatrix}$. Let's treat $r$ as the input in our algorithm. Without appealing to the semiprimitivity or artinian conditions, the only constructable elements are integer polynomials in $r$. (Is this correct?) Thus, we have access to $0,1,r,1+r$, and nothing else (since $2=0$ and $r^2=0$; although we do know these two equalities).

The left ideals generated by these elements are $(0)\subsetneq Rr\subsetneq R$, and the right ideals are $(0)\subsetneq rR\subsetneq R$ (and these containments are provable with what we constructively know). Thus, appealing the the left and right artinian conditions will not yield anything.

The element $r$ is nilpotent. Since the Jacobson radical is zero, we know that some multiple of $r$ must not be nilpotent. But do we have access to any constructible element $s$ such that $rs$ is not nilpotent? Does the zero Jacobson radical hypothesis allow us any constructible element whatsoever?

So I guess my ultimate question is: Are there any further elements can we construct in this scenario? If not, then note we cannot construct an inner inverse for $r$.

I'm not as familiar with constructive proofs, so this "answer" is really just a couple questions.

Consider the ring $R=\mathbb{M}_2(\mathbb{F}_2)$, and the element $r=\begin{pmatrix} 0 & 1\\ 0 & 0\end{pmatrix}$. Let's treat $r$ as the input in our algorithm. Without appealing to the semiprimitivity or artinian conditions, the only constructable elements are integer polynomials in $r$. (Is this correct?) Thus, we have access to $0,1,r,1+r$, and nothing else (since $2=0$ and $r^2=0$; although we do know these two equalities).

The left ideals generated by these elements are $(0)\subsetneq Rr\subsetneq R$, and the right ideals are $(0)\subsetneq rR\subsetneq R$ (and these containments are provable with what we constructively know). Thus, appealing the the left and right artinian conditions will not yield anything.

The element $r$ is nilpotent. Since the Jacobson radical is zero, we know that some multiple of $r$ must not be nilpotent. But do we have access to any constructible element $s$ such that $rs$ is not nilpotent? What kind of elements does a zero Jacobson radical hypothesis allow us claim existence for?