On hearts which are module categories
Given a torsion pair $\mathbf{t} = (\mathcal{T} ;\mathcal{F})$ in a module category $R$-Mod we give necessary and sufficient conditions for the associated Happel-Reiten-Smal\o $\text{ }$ t-structure in $\mathcal{D}(R)$ to have a heart $\mathcal{H}_{\mathbf{t}}$ 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 $\mathbf{t}$ for which $\mathcal{H}_{\mathbf{t}}$ is a module category in the following cases: i) when $\mathbf{t}$ is the left constituent of a TTF triple, showing that $\mathbf{t}$ need not be HKM; ii) when $\mathbf{t}$ is faithful; iii) when $\mathbf{t}$ 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