Very incorrect actually, you are misinformed like many others.
While it is true that all species of carp are "introduced" fish not native to the United States, only a few species are considered "invasive", and detrimental to an ecosystem. Particularly the Silver Carp, Bighead Carp, and Black Carp can be damaging, and in some cases the Grass Carp can be as well. Often the grass carp you see in ponds are completely legal, sterilized fish, unable to reproduce but kept in the ponds to control vegetation.
The Bighead, Black, and Silver Carp have not made their way to Ohio yet, and are found nearly exclusively in the Mississippi River Basin and it's tributaries. The Common Carp in particular is an introduced European species, and while it is very prevalent it is seldom damaging to its habitat. They are quite fun to catch as well, as they truly bolt when they are hooked and put up quite a fight.