Neutrons don't decay while being part of a stable atom, because the atom has actually less energy than the sum of the constituents -- the difference is the binding energy. Look at deuterium, for example. It has a mass of 2.0141 u. A proton alone is 1.0073 u, and a neutron is 1.0087 u. Deuterium is lighter than the mass of proton + neutron. It's also slightly lighter than two protons, so the neutron cannot decay without external energy input.
This actually brings me to a physics question I've had for a while and if it is unrelated, please feel free to ignore it as I might be mixing concepts.
Both fusion and fission release energy when they occur. Which seems somewhat weird to me. Is it the cases that the reason a stable atom has less energy than the sum of its parts (as you pointed out) is because it gave off some energy during the fusion process?
Pretty much. To simplify a bit: the most stable atom is iron-56 anything lighter can be fused and anything heavier can be split to release energy. Essentially after iron the forces that bind atoms together start to lose out to the repulsion between it's constituents, which makes heavier atoms more and more unstable.
This is also why stars that start to fuse iron together will start to cool down.
Edit:
If neutron is likely to decay into. Why *neutrons* dont decay while being part of an atom?
(Wouldn’t this be example of a structure that prevents decaying?)