The way that ETS asks question 66 on Practice Test GR0568 actually answers your question:

https://www.mathsub.com/gr0568-66/First of all, the question starts off saying "Let

be a ring with multiplicative identity". The fact that the existence of a multiplicative identity is explicitly stated implies that the definition of a ring used by the ETS does

not necessarily include the identity, even though there are some textbook authors that do require it.

Second of all, choice I says "

is commutative". The fact that this is also explicitly stated means that the ETS's definition of a ring also does

not necessarily include commutativity. Again, some authors differ on this, though that's often because you need commutativity to be able to do things (anything that falls under the umbrella of "commutative algebra").

So, all in all, the definition of a ring you should use is that

forms an abelian group under addition, is closed and associative under multiplication, and satisfies the distributive property of multiplication over addition. That's it --- any other properties may or may not be true depending on the question, and will be explicitly stated when necessary.

This is highly reminiscent of how the College Board structures its questions for the AP Calculus test. For example, since textbooks often differ in whether the intervals of increase and decrease for a function should be open or closed intervals, they're very explicit with their language and ask questions such as "Find the open intervals on which

is increasing". They do not ask questions that could be ambiguous, because that would decrease the reliability of their test.

Hope this helps.