  We develop a general formalism to study laser operation in active micro-ring
resonators supporting two counterpropagating modes. Our formalism is based on
the coupled-mode equations of motion for the field amplitudes in the two
counterpropagating modes and a linearized analysis of the small perturbations
around the steady state. We show that the devices including an additional
S-shaped waveguide establishing an unidirectional coupling between both modes
-- the so-called Taiji resonators (TJR) -- feature a preferred chirality on the
laser emission and can ultimately lead to unidirectional lasing even in the
presence of sizable backscattering. The efficiency of this mode selection
process is further reinforced by the Kerr nonlinearity of the material. This
stable unidirectional laser operation can be seen as an effective breaking of
$\mathcal{T}$-reversal symmetry dynamically induced by the breaking of the
$\mathcal{P}$-symmetry of the underlying device geometry. This mechanism
appears as a promising building block to ensure non-reciprocal behaviors in
integrated photonic networks and topological lasers without the need for
magnetic elements.
