Ring modulation involves multiplying two signals together, resulting in a spectrum composed of the sum and difference frequencies of the two waveforms.
If there are frequency components in one waveform and frequency components in a second waveform, as many as components will be present in the spectrum of the waveform produced by multiplication of the two.
By carefully selecting the relationship between carrier and modulating frequencies, interesting and rich spectra can be produced via a relatively inexpensive computational operation.