Battery charging light is on even when power adapter unplugged

I have a Thinkpad T41. Yellow light for battery charging does not go even after switching the laptop off and power jack is disconnected. Battery does not charge at all with power source connected.
Can you please help?

First of all, try reconnection the battery if you haven’t done it already.
The battery plugs directly into the connector attached to the motherboard. If the battery charging light stay on even when the power adapter is unplugged and the laptop is switched off, it’s either bad battery or something is wrong with the motherboard. The only way to find the culprit is testing the laptop with another known good battery.
Also, you can try this. Remove the battery and plug in the power adapter. If the battery charging LED lights up even when the battery is removed, most likely you have a problem with the motherboard.