  A simplified but very instructive analysis of the seesaw mechanism is here
performed. Assuming a nearly diagonal Dirac neutrino mass matrix, we study the
forms of the Majorana mass matrix of right-handed neutrinos, which reproduce
the effective mass matrix of left-handed neutrinos. As a further step, the
important effect of a non diagonal Dirac neutrino mass matrix is explored. The
corresponding implications for the baryogenesis via leptogenesis and for the
neutrinoless double beta decay are reviewed. We propose two distinct models
where the baryon asymmetry is enhanced.
