A study is made of the effect of a temperature‐dependent Coulomb‐like resistivity on the planar tearing mode. The local evolution of the temperature is described by an energy equation which includes Joule heating and optically thin radiation. The resulting system of coupled linear magnetohydrodynamic equations is solved numerically, and eigenfunctions and growth rates are obtained. In the absence of radiation, there are two distinct solutions above a critical value of the magnetic Reynolds numberS, a tearing‐like mode and a Joule‐heating mode. Below this point, the growth rates coalesce into a conjugate‐complex pair. When stable radiation (dR/dT>0) is added, the heating mode disappears and a modified tearing excitation exists to much lower values ofSbefore its growth is cut off by Ohmic heating. Examples are given for solar coronal parameters, and for those characteristic of fusion‐research devices. The introduction of an effective value for the resistivity, in the presence of energy transport, allows a simple qualitative discussion of the different modes.