Orbits of Free Cyclic Submodules over Rings of Lower Triangular Matrices

Given a ring $T_n,n⩾2$, of lower triangular $n\times n$ matrices with entries from an arbitrary field $F$, a complete description is performed of the orbits of free cyclic submodules of ${}^2{T}_n$, under the action of the general linear group $G{L}_2(T_n)$. It is given its total number, which is equal to the Bell number $B_n$, and there are shown their representatives.