How do I know the definition of rings or of anything on the GRE given that definitions can vary?

Question

How do I know the definition of rings or of anything on the GRE given that definitions can vary? :|

Context is rings:

1. GRE 0568 #66: On whether or not exactly 2 right ideals give a non-commutative field and related questions

   • A ring with exactly 2 ideals is a field and hence commutative...IF the ring is commutative and thus there are no such notions of right or left ideals I guess.

2. GRE 9768 #60 Boolean rings: 1. Does $(s+t)^2=s^2+t^2$ imply $s+s=0$? 2. Idempotent matrices do not form a ring?

   • I might have incorrectly argued $(s+t)^2=s^2+t^2$ implies $s+s=0$ because I assumed the ring contains $1$.

3. Do subrings contain 0, the additive identity because $1-1=0$ in subrings as in subfields?

   • If subrings contain 0, then I hope to rule out subsets as subrings if they do not contain 0. I believe this will help me work more quickly in the exam. I don't know if subrings still contain 0 under a different definition of rings. Even if the definition of rings is the same, how do I know the definition of subrings is still the same?

But even outside rings, how do I know that the GRE has the same definition for fields, holomorphic/analytic functions, Hausdorff spaces, uniform continuity or convergence or even rectangles?!

Answer 1

If one considers Princeton GRE to be canon, then rings are not necessarily commutative and do not necessarily have identity elements.

ETA: Hopefully rings are associative in multiplication. I read on mathematicsgre.com that some authors don't assume this.