We investigate the rectification characteristics of a carbon nanotube (CNT) ring transistor contacted by CNT leads, whose novel properties have been recently communicated by Watanabe et al. [Appl. Phys. Lett. 78, 2928 (2001)]. Our numerical results are validated by the experimental observations. Moreover, we report on additional features of the transmission of the CNT ring device which are associated with the possibility of breaking the lead inversion symmetry. The linear conductance displays a ``chessboard''-like behavior alternated with anomalous zero-lines which should be directly observable in experiments. By studying a one-dimensional ring displaying broken inversion-symmetry as a simplified version of this system, which allows for analytical treatment, we are able to discriminate structural properties (quasi-onedimensional confinement) from pure topological effects (ring configuration). In the simplified system, jumps in the transmittance phase occur in correspondence to both (a) the zeros in the transmission at the eigenstates of the isolated ring (which lead to a discrete set of singular states in the density of states) and (b) the destructive interference events.