  A ring $R$ is said to be i-reversible if for every $a,b$ $\in$ $R$, $ab$ is a
non-zero idempotent implies $ba$ is an idempotent. It is known that the rings
$M_n(R)$ and $T_n(R)$ (the ring of all upper triangular matrices over $R$) are
not i-reversible for $n \geq 3$. In this article, we provide a non-trivial
i-reversible subring of $M_n(R)$ when $n \geq 3$ and $R$ has only trivial
idempotents. We further provide a maximal i-reversible subring of $T_n(R)$ for
each $n\geq 3$, if $R$ is a field. We then give conditions for i-reversibility
of Trivial, Dorroh and Nagata extensions. Finally, we give some independent
sufficient conditions for i-reversibility of polynomial rings, and more
generally, of skew polynomial rings.
