Yes, the whole point is that it is a cycle. The fields don’t affect by much the total number of carbon atoms in the atmosphere. Without the fields, other plants would grow there, then die, decay, and release carbon into atmosphere.

At best, growing rice and flooding fields might affect the balance between the amount of methane and amount of CO2 in the steady state, but this is of no practical consequence, because the methane will not accumulate year over year, like the emissions from fossil fuels do.

Suppose we grow twice as much rice as we do now. This means that the steady state will have twice as much methane coming from the fields as it has now. Hell, let’s actually assume that all existing methane in atmosphere is from rice fields, and actually increase amount of methane in atmosphere by a factor of 2. What is the impact to the climate, according to our current models? Well, this is equivalent to less than 10% increase in total CO2 in the atmosphere. That’s like a difference between 2000 and 2020, which is to say, almost unobservable — and remember I assumed we increase total methane by factor of 2, not just methane from the fields.