Self-gravity becomes competitive as an angular momentum transport process in accretion discs at large radii, where the temperature is low enough that external irradiation likely contributes to the thermal balance. Irradiation is known to weaken the strength of disc self-gravity, and can suppress it entirely if the disc is maintained above the threshold for linear instability. However, its impact on the susceptibility of the disc to fragmentation is less clear. We use two-dimensional numerical simulations to investigate the evolution of self-gravitating discs as a function of the local cooling time and strength of irradiation. In the regime where the disc does not fragment, we show that local thermal equilibrium continues to determine the stress - which can be represented as an effective viscous alpha - out to very long cooling times (at least 240 dynamical times). In this regime, the power spectrum of the perturbations is uniquely set by the effective viscous alpha and not by the cooling rate. Fragmentation occurs for cooling times tau < beta_crit / Omega, where beta_crit is a weak function of the level of irradiation. We find that beta_crit declines by approximately a factor of two, as irradiation is increased from zero up to the level where instability is almost quenched. The numerical results imply that irradiation cannot generally avert fragmentation of self-gravitating discs at large radii; if other angular momentum transport sources are weak mass will build up until self-gravity sets in, and fragmentation will ensue.
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