Measurement of the chemical and isotopic composition of cosmic rays is essential for the precise understanding of their propagation in the galaxy. While the model parameters are mainly determined using the B/C ratio, the study of extended sets of ratios can provide stronger constraints on the propagation models. In this paper, the relative abundances of light-nuclei lithium, beryllium, boron, and carbon are presented. The secondary-to-primary ratios Li/C, Be/C, and B/C have been measured in the kinetic energy range 0.35-45 GeV nucleon(-1). The isotopic ratio Li-7/Li-6 is also determined in the magnetic rigidity interval 2.5-6.3 GV. The secondary-to-secondary ratios Li/Be, Li/B, and Be/B are also reported. These measurements are based on the data collected by the Alpha Magnetic Spectrometer AMS-01 during the STS-91 space shuttle flight in 1998 June. Our experimental results are in substantial agreement with other measurements, where they exist. We describe our light-nuclei data with a diffusive-reacceleration model. A 10%-15% overproduction of Be is found in the model predictions and can be attributed to uncertainties in the production cross-section data.