I know it does not make sense that if a subring of a ring $R$ is commutative, then $R$ is also commutative. (For example, the set consisting of the matrices whose all entries except (1,1)-entry are zero, is a subring of ring of 2x2 real matrices.)
I also considered the case of a ring containing a subring with identity, but I had no ideas. Maybe it seems that it does not make sense, either.
Who give me some examples supporting my guess?