Semicommutative Subrings of Matrix Rings
A ring $R$ is called semicommutative if for every $a\in R$, $r_R(a)$ is an ideal of $R$. It is well-known that the $n$ by $n$ upper triangular matrix ring is not semicommutative for any ring $R$ with identity when $n\geq 2$. We show that a special subring of upper triangular matrix ring over a reduced ring is semicommutative.