According to the Concise Oxford Dictionary (1964) one of my very favourite books, RIVE means to Rend, cleave, wrench away or off or from, strike asunder. Split. Make by splitting.
Surprisingly, it is the root of the word RIVER.
Gives you a whole different idea of what a river once was thought to be. As if the bed of the river, and the difficulties of movement through a landscape made up of RIVEN-apart mountains, were more important than the streams forming those landforms.
Adding RIVER to various words we get river-horse, riverside, riverless, riverboat.
RIVULET and RIVERAIN or RIVERINE are derived from RIVER. Second generation words as it were. Though RIVULET, a small river, may derive more directly from the Latin RIVUS.
REAVE and REIVE (committing ravages, carrying off by force, forcibly depriving etc etc) are in some dictionaries held to derive from RIVE. Making a delicious irony of the REAVERS in the film Serenity slaughtered by the female fighting machine, RIVER.
Alas, it is not to be. Or rather, it is incorrectly so. According to Oxford, anyway. Oxford holds that REAVE and REIVE are the children of a different parentage entirely.
Note, too, the word DERIVE (obtain from a source) that I used to describe the various ancestries. Its progenitor is the same as RIVULET, the Latin word RIVUS for stream.