The vital ingredient of household smoke detectors is a very small quantityc of Am-241 as americium dioxide (AmO2).
Americium-241 emits alpha particles and low energy gamma raysd. The alpha particles emitted by the Am-241 collide with the oxygen and nitrogen in air in the detector's ionisation chamber to produce charged particles (ions). A low-level electric voltage applied across the chamber is used to collect these ions, causing a steady small electric current to flow between two electrodes. When smoke enters the space between the electrodes, the smoke particles attach to the charged ions, neutralizing them. This causes the number of ions present – and therefore the electric current – to fall, which sets off an alarm.
The radiation dose to the occupants of a house from a domestic smoke detector is essentially zero, and in any case very much less than that from natural background radiation. The alpha particles are absorbed within the detector, while most of the gamma rays escape harmlessly. The small amount of radioactive material that is used in these detectors is not a health hazard and individual units can be disposed of in normal household wastee.
Even swallowing the radioactive material from a smoke detector would not lead to significant internal absorption of Am-241. Americium dioxide is insoluble, so will pass through the digestive tract without delivering a significant radiation dose. (Americium-241 is however a potentially dangerous isotope if it is taken into the body in soluble form. It decays by both alpha activity and gamma emissions and it would concentrate in the skeleton).