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A class inherits all the interfaces implemented by any of its base classes. This is basic Java.
It will have an "IS-A" relationship with Serializable. That is, anywhere a Serializable is expected, it will be syntactically legal to use a B. However, if you have non-transient, not-static, non-Serializable member variables in B (or in A for that matter), then serialization will fail at runtime, so in that sense, it is not serializable (with a lowercase "s", as you wrote it--that is, not able to be serialized).