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?!

  • $\begingroup$ The definitions to which you refer are all pretty standard definitions... $\endgroup$ – Xander Henderson Oct 23 '18 at 4:00
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    $\begingroup$ It's about logical equivalence or restricting to certain classes of objects. The GRE will be clear enough to say a ring with unity or without etc... Otherwise that's just bad math. I understand you're confusion about rings but you are making too big deal out of it. They aren't different definitions. We just restrict our attention to certain types of rings sometimes. Like commutative, rings with unity etc. Again, any problem will make it very clear what kind of ring you are dealing with for example $\endgroup$ – RhythmInk Oct 23 '18 at 4:05
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    $\begingroup$ In all of these questions, the properties you need have been stated. For example, in the one about two ideals, it is stated that the ring has a unit. For the question about Boolean rings, this has been left unspecified, but a correct answer to the question will not need to assume the ring has a unit. $\endgroup$ – Joppy Oct 23 '18 at 4:41
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    $\begingroup$ Unital is a common assumption, but not everyone assumes this - obviously whoever set these exams does not take unital in their definition of a ring. Also - welcome to the world of algebra where people often neglect to define their basic assumptions :P It’s quite annoying. $\endgroup$ – Joppy Oct 23 '18 at 4:50
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    $\begingroup$ @BCLC If everyone defined everything at the start of their papers, no-one would get anywhere since every paper would start off with a textbook. Good authors will say things like "Associative commutative ring (not necessarily unital)" or something along those lines to clarify what they mean. $\endgroup$ – Joppy Oct 23 '18 at 5:32

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.


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