Not necessarily cow farts, but manure, fertilizer, and landscaping.
Methane comes primarily from livestock digestion (known as enteric fermentation) and the way livestock manure is managed. It contributes the most to agricultural emissions of greenhouse gases.
The second largest contributor is nitrous oxide, which results mostly from agricultural fertilizer application to soils and from manure management.
Carbon dioxide emissions come from increased decomposition of plant matter in soils and from converting lands to agricultural uses. Those emissions are partially offset by the increased plant matter stored in cropland soils.
You’re not wrong about the same gasses being created by decomposing grass and digested grass, but like most things, it’s a multifaceted issue.
As they say, you can’t get snakes from chicken eggs.
Truth be told the nugget about the same gasses being released from digestion and decomposition was news to me, so thank you for that. My knee-jerk reaction was to refute it but I realized that I truly didn’t know for sure. So I checked, lo and behold, I was wrong, and now I’ve learned something today.