  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
