I want to compute the Jacobson radical, the right socle, and the left socle of the ring of $3\times 3$ upper triangular matrices with entries in $\mathbb Z_4$ and such that the entries on the main diagonal are equal.
I know that the Jacobson radical of the ring of $3\times 3$ upper triangular matrices is comprised of those with zero main diagonal, and that, when the ring $R$ is Artinian, the right (left) socle of $R$ is the left (right) annihilator of the Jacobson radical of $R$. Thanks for any suggestion and/or help!
Well, you have that $\begin{bmatrix}0&b&c\\ 0& 0& d\\ 0&0&0\end{bmatrix}$ is a nilpotent ideal, and that $\begin{bmatrix}2&0&0\\ 0& 2& 0\\ 0&0&2\end{bmatrix}$ is a central nilpotent, so the Jacobson radical must contain at least $\begin{bmatrix}2a&b&c\\ 0& 2a& d\\ 0&0&2a\end{bmatrix}$ for any choice of $a,b,c\in \mathbb Z_4$.
The quotient by this ideal is exactly $\mathbb Z_2$, so apparently we have already found the Jacobson radical.
I trust you can compute the annihilators after knowing this? Both socles have just four elements apiece.
That is not really correct. If you have $R$ and take the quotient in $T_n(R)$ by the strictly upper diagonal matrices, you get $R^n$, which is not Jacobson semisimple unless $R$ already was. So the radical is strictly bigger than the strictly upper triangular matrices, and contains some elements on the main diagonal (namely the elements in $J(R)$.