Russia imposes cyrillic alphabet in other languages of its empire

Hmmm, there had been some attempts at introducing pre-Revolutionary Cyrillic script, but not on the basis of Russification. Rather, if we look at such literary languages for, say, Chuvash, or Kazakh, or Aleut, or Kerashen Tatar, for example, the sole reason was to expand the Orthodox Church by converting minority populations. One could stil speak (and hence now read and write) his or her native language as long as s/he worships in Church Slavonic along with everyone else. So it's been done, though in the case of Kazakh it actually took the Soviet period to make it stick (with some judicious revision to make it match the modern Russian alphabet).
 
Well, their languages were written in the Arabic script

But, AFAIK, they were not forced to write in Russian unless it was a matter of the official communications. The Cyrillic-based alphabets for the nations that were traditionally using the Arabic script had been, IIRC, invented and enforced only in the Soviet times. Notice that both Georgia and Armenia preserved their alphabets even under the Soviets.
 
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