Some teachers went to Russia to be educated in fine arts

I'm not quiet sure that I understood your question, but anyhow I would say
Some teachers went to Russia to study fine arts. if I didn't answer you question please clarify your request

Thank you that you give me an actual activity to touch it.

"in" shows a matrix in which some teachers went to Russia to be educated.

Give new example again.
Children should be educated in a correct way.
rephrased: A correct way in which children should be educated.

bepleased"in" shows a matrix in which some teachers went to Russia to be educated.

I'd say not. "Educated in" is equivalent to "taught".

Give new example again.
Children should be educated in a correct way.
rephrased: A correct way in which chi

As your suggestion, the "in" has two logic meanings.

In first sentence, it shows the thing of "some teachers were educated" attends on the purpose of fine arts.

In second sentence, it shows the thing of "a correct way" as the subject or activity in which "children should be educated" made.

Give your help again please.

Your English is hard to follow, but I think you have it right.