Yes you can, Patricia and I have done this with success! It is better if you can pump the water into a growing container above the tank that is filled with gravel or clay balls (hydroton) and then let it drip back into the tank. You have to be careful with the pump though to make sure you have an overflow safety in case the growing container gets clogged so you don’t have a flood, but there are ways to do this. But as you said, you can just let the plants’ roots grow directly into the tank (no pump involved). With this the problem is that not every plant likes to have roots that wet, but some do. Other issues I have run into are that you need to make sure your fish aren’t the kind that eat the roots (my cichlids that I have right now won’t let a plant of any kind grow for 1 day without devouring them! Another issue is light for the plants. Usually the top of aquariums are not near a direct light source, so you have to use daylight compact fluorescent bulbs to get enough lumens to the plant. A couple of CFLs set to a timer with 12 hours on and twelve hours off works well. Set the timer to less light if you want the plant to flower, simulating the end of the growing season.
Water & Air Pumps | Aquarium Basics | Tropical Fish Hobbyist Magazine
A fish tank filter is meant to ensure a healthy, clean environment for your fish as well as maintain a presentable, sparkly appearance of your aquarium. The AquaClear 20 Power Filter ensures just that by incorporating a multi-stage filtration system. This filter is easy to install and comes with an energy efficient pump. It’s ideal for small fish tanks and can handle a capacity of 20 gallons. The AquaClear foam that comes with this filter acts as a sponge that traps dirt while the activated carbon does a good job of treating any chemicals that may be present in the water.
Aquarium Help and Answers | Tetra Aquarium - Tetra-Fish
These are placed below gravel from where they suck water to trap debris then pass the cleaned water through tubes back to the tank. An air pump and air stone are placed at the top of the tube so as to create suction. Under gravel fish tank filters aren’t recommended because they allow for accumulation of debris at the bottom of the aquarium leading to increased levels of hydrogen sulfide, which is harmful to fish.
Do not clean the aquarium with soapy water or with cleaning products