Small fish tank question?

I have a female betta that was in a small bowl for about 3 weeks and I noticed it was not happy. So I went out to Petsupermarket and bought a one gallon fish tank with a lighted top, air pump, under-gravel "filter", along with some rocks, and ornament and a plant. I am asking if the under-gravel filter is enough to keep the tank clean??? My fish poops alot and I try to feed her less but she gets to hungry and I notice this because she attacks the food when I dont feed her for an amount of time. Also if the under-gravel filter is no good what are some REALLY small tank filters I can buy. -Thank you
I had my old betta for 3 years without a heater
 
this is the best filter for 1 gallon tank i have this like for 5 years and still working fine..
http://www.petco.com/product/12148/Red-Sea-Deco-Art-Nano-Filter.aspx?CoreCat=LN_Shopping_FishSupplies_Filters
 
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