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Given a ring $$T_n\ (n\geqslant 2)$$ T n ( n ⩾ 2 ) of lower triangular $$n\times n$$ n × n matrices with entries from an arbitrary field F , we completely classify the orbits of free cyclic submodules of $$^2T_n$$ 2 T n under the action of the general linear group $$GL_2(T_n)$$ G L 2 ( T n ) . Interestingly, the total number of such orbits is found to be equal to the Bell number $$B_n$$ B n . A representative of each orbit is also given.

Orbits of Free Cyclic Submodules Over Rings of Lower Triangular Matrices