The proof of Proposition 1.9 of this paper http://ac.els-cdn.com/0168007294900221/1-s2.0-0168007294900221-main.pdf?_tid=5d45c892-b816-11e6-8401-00000aab0f27&acdnat=1480631867_bccc7147520bd9d7356358bb364ec8e1 by Belgradek says the answer is no. 

However Theorem 1.13 says the answer is yes if the rings are commutative.