The equation provided is:
\[ ^{146}\text{C} \rightarrow ^{147}\text{N} + 0^{-1}\text{e} \]
In this equation, a carbon-146 nucleus undergoes beta-minus decay, as indicated by the emission of a beta particle (which is an electron, denoted as \(0^{-1}\text{e}\)). During beta-minus decay, a neutron in the nucleus is transformed into a proton, resulting in an increase in the atomic number (from 6 for carbon to 7 for nitrogen) while the mass number remains unchanged.
Therefore, the response is:
beta-minus decay