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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?!

YuiTo Cheng
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BCLC
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  • The definitions to which you refer are all pretty standard definitions... – Xander Henderson Oct 23 '18 at 04:00
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    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 – RhythmInk Oct 23 '18 at 04:05
  • @RhythmInk In 9768 #60, is $1$ in the ring? In 0568, is $R$ not necessarily commutative just because there's a choice that says commutative? :| – BCLC Oct 23 '18 at 04:30
  • @XanderHenderson Algebra by Michael Artin defines rings to be commutative and having multiplicative identity $1$. Apparently, these may not be the same definitions as in the GRE, so I can't necessarily apply the propositions I learned in the book. What do I do please? – BCLC Oct 23 '18 at 04:31
  • Is the identity matrix idempotent? – RhythmInk Oct 23 '18 at 04:38
  • @RhythmInk ...Yes? Then again I think you're going to say something about a definition of unital rings vs non-unital rings. – BCLC Oct 23 '18 at 04:41
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    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. – Joppy Oct 23 '18 at 04:41
  • @Joppy Hmmm...sooo basically I should assume GRE rings are defined as non-commutative and without multiplicative identities? – BCLC Oct 23 '18 at 04:44
  • I think that everyone will agree that a ring is an abelian group equipped with a bilinear associative multiplication. Unital is a very common assumption also. But I would never assume that a ring was commutative until someone told me. – Joppy Oct 23 '18 at 04:47
  • @Joppy My question is about the GRE in relation to your statement 'In all of these questions, the properties you need have been stated.' If rings are unital, then why did 0568 #66 say such? :| – BCLC Oct 23 '18 at 04:48
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    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. – Joppy Oct 23 '18 at 04:50
  • @Joppy 'Also - welcome to the world of algebra where people often neglect to define their basic assumptions :P It’s quite annoying.' Actually, so, what some maths research papers actually don't define rings from at the start? How do journals accept this??!!!! – BCLC Oct 23 '18 at 05:31
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    @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. – Joppy Oct 23 '18 at 05:32
  • @Joppy Ok fine. thanks. lol. – BCLC Oct 23 '18 at 05:33
  • @Joppy you can still post as answer... – BCLC Jan 17 '21 at 23:28

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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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