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It seems that ETS has never given any standard references for the terminologies that are used in the GRE Sub Mathematics exam. While in the Cracking the GRE Mathematics Subject Test, a ring has been defined as not requiring the existence of a multiplicative identity, i.e. the unity, which affects a number of related concepts too, the subring, the ring homomorphism etc. However, in most algebra books (for instance, Jacobson's), such an object is given the name rng, and the ring is always assumed to have a unity. So what's the ETS's definition of a ring?
It's incredibly unlikely that the ETS will ask a question that banks on whether the definition of a ring contains a unity element or not, precisely because different texts and different professors give different definitions -- they will always be clear about what they're asking. This is similar to how on the AP Calculus test, there is never a question whose crux is whether you consider the limit of x^-2 as x approaches 0 to be "does not exist" or "infinity".