On hearts which are module categories

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Given a torsion pair t = (T;F) in a module category R-Mod we give necessary and sufficient conditions for the associated Happel-Reiten-Smalø t-structure in D(R) to have a heart Hₜ which is a module category. We also study when such a pair is given by a 2-term complex of projective modules in the way described by Hoshino-Kato-Miyachi ([HKM]). Among other consequences, we completely identify the hereditary torsion pairs t for which Hₜ is a module category in the following cases: i) when t is the left constituent of a TTF triple, showing that t need not be HKM; ii) when t is faithful; iii) when t is arbitrary and the ring R is either commutative, semi-hereditary, local, perfect or Artinian. We also give a systematic way of constructing non-tilting torsion pairs for which the heart is a module category generated by a stalk complex at zero