We use the W_Ha versus [NII]/Ha (WHAN) diagram to provide a comprehensive emission-line classification of SDSS galaxies. This classification is able to cope with the large population of weak line galaxies that do not appear in traditional diagrams due to a lack of some of the diagnostic lines. A further advantage of the WHAN diagram is to allow the differentiation between two very distinct classes that overlap in the LINER region of traditional diagnostic diagrams. These are galaxies hosting a weakly active nucleus (wAGN) and retired galaxies (RGs), i.e. galaxies that have stopped forming stars and are ionized by their hot evolved low-mass stars. A useful criterion to distinguish true from fake AGN (i.e. the RGs) is the ratio (xi) of the extinction-corrected L_Ha with respect to the Ha luminosity expected from photoionization by stellar populations older than 100 Myr. This ratio follows a markedly bimodal distribution, with a xi >> 1 population composed by systems undergoing star-formation and/or nuclear activity, and a peak at xi ~ 1 corresponding to the prediction of the RG model. We base our classification scheme on the equivalent width of Ha, an excellent observational proxy for xi. Based on the bimodal distribution of W_Ha, we set the division between wAGN and RGs at W_Ha = 3 A. Five classes of galaxies are identified within the WHAN diagram: (a) Pure star forming galaxies: log [NII]/Ha < -0.4 and W_Ha > 3 A. (b) Strong AGN (i.e., Seyferts): log [NII]/Ha > -0.4 and W_Ha > 6 A. (c) Weak AGN: log [NII]/Ha > -0.4 and W_Ha between 3 and 6 A. (d) RGs: W_Ha < 3 A. (e) Passive galaxies (actually, line-less galaxies): W_Ha and W_[NII] < 0.5 A. A comparative analysis of star formation histories and of other properties in these different classes of galaxies corroborates our proposed differentiation between RGs and weak AGN in the LINER-like family. (Abridged)
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