Does decomposing add nutrients to the soil?

Yes. Decomposition may take place aerobically, with the presence of oxygen; or anaerobically, in its absence. Anaerobic decomposition produces methane, which is a greenhouse gas. And methane contributes to the disagreeable smell of landfills. Aerobic decomposition needs adequate air, moisture and temperature levels. With those levels properly reached, dead and decaying materials break down and contribute to the necessary amounts and ranges of the 16-17 nutrients that are necessary for healthy plants and soils. For compostable materials may be grouped into the green nitrogen-rich category, or the brown carbon-rich category. Both carbon and nitrogen are included in the list of 16-17 necessary nutrients.