Smoke detector - Smoke detector connecting battery and power supply under one socket?

Hello.
I would like to install smoke detectors in my house. I am interested in powering them from a 9V mains supply with a battery as additional protection in case of power failure. Such detectors cost more than 100PLN.
My question is this. Can I plug the power supply and a normal battery into the battery socket? The battery so that it is not charged would be behind the semiconductor diode.
Would such a solution work properly?
If so, which diode should I choose? Give one or two (for each pole?)
Thank you in advance for your reply.

You can do that. You only need one diode on the plus side of the battery. You can use an ordinary 1A rectifying diode such as 1N4001.

You can't!
The power supply must give a voltage higher than the battery voltage and must not draw current from the output when switched off otherwise the battery will be quickly discharged by the power supply in the absence of power.
The power supply also through a diode or it must already have one built into the output.