Transpose of an Ideal in a Matrix Ring I just recently proved that if $J \subseteq M_n(R)$ is a left ideal, where $R$ is some ring, then $J^T := \{A^T \mid A \in J \}$ is a right ideal, and vice-versa. This got me wondering: is there some necessary and sufficient conditions involving a nonempty subset $J$ being an ideal and $J^T = J$, etc.? 
If $J$ is an ideal, then it is both a left and right ideal, and so $J^T$ is both a right and left ideal (according to what I proved), which means it is just an ideal proper. But I am not sure how to conclude from $J$ being an ideal that $J^T = J$; I am very doubtful that it is possible. Perhaps there are additional stipulations that will guarantee this.  

A two-sided ideal of $M_n(R)$ has the form $M_n(I)$ where $I$ is a two-sided ideal of $R$. For clarity, $M_n(I)$ denotes the set of matrices with entries in $I$.
This is a well-known result on matrix rings, you can find it in every textbook (maybe as an exercise).
If $J$ is a right or left ideal and $J=J^T$, then $J$ is a two-sided ideal, so $J=M_n(I)$ for some two-sided ideal $I$ of $R$. Conversely, if $I$ is a two-sided ideal of $R$, then $M_n(I)^T=M_n(I)$.
