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You are right. It's a very common name, used in Russia for a thousand years or so. ;)
Though it's spelled (and pronounced) Дмитрий in Russian - there's no sound "ee" after first "d". For some...

Alicia Silverstone = Алисия not Алисия nor Алиша[/quote]

That's because her name is pronounced Алисия. The name Alicia can be pronounced in a number of ways, so the person should choose the...

Then it must be Алиша Бэмман.[/quote]

Я согласен.

Bemman is German / Jewish.

Да, обычно сохранят суффикс -ман (с буквы А, а не Э).