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

Posted: Tue Oct 23, 2018 12:34 am
Definition of ring varies: https://math.stackexchange.com/questions/48587

It may or may not be commutative.

It may or may not have a multiplicative identity.

What is the definition of a ring for the GRE please? What about fields, holomorphic/analytic functions, Hausdorff spaces, uniform continuity or convergence or even rectangles?!

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How do I know the definition of rings or of anything on the GRE given that definitions can vary?
https://math.stackexchange.com/questions/2967070

Whether Euclid considered squares to be rectangles
https://math.stackexchange.com/questions/2702928

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

Posted: Tue Oct 23, 2018 8:25 am
These definitions aren't ambiguous. For example any ring, on the GRE or not, must satisfy the following axioms:
1. It is an abelian group under addition
2. Closed under multiplication
3. Associative under multiplication
4. Obeys the distribution law under multiplication

Now if it has a multiplicative identity, it's known as a ring with identity, if it commutes under multiplication then it's known as a commutative ring. Unless specifically stated (or proven), the above two conditions cannot be assumed.

Similarly, you can find the definitions for any of the other terms you're looking for either online or in a textbook. The beauty of math is, in part, it's rigor in definitions/theorems leaving no room for ambiguity.

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

Posted: Tue Oct 23, 2018 8:57 am
The question applies, though. I studied in Uruguay, and the definition of rings we worked with was the one you mentioned. But the definition of increasing function was different. I was discussing with a coworker of mine, and we would never get to an agreement till we realised that there are 3 different ways to define it, and they are not equivalent.

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

Posted: Tue Oct 23, 2018 9:21 am
I never said ambiguous. I was just want to know what the definition of a ring is in the GRE. Technically the ring definition in the GRE *IS* different from the one in Artin right? This has a few implications. One is that I can't use some of the properties that rings in the Artin book have like how exactly 2 ideals implies field because that property assumes commutativity of rings. Another is the definition of a subring. It's not confusing or ambiguous at all. It's just not explicitly given by ETS.

And that's just for rings! How should I know that what I understand to be 'groups' is what they consider to be 'groups' ? What if ETS' groups are abelian or something? What if ETS' rectangles are actually exclusive rectangles (exclusive of squares) ?

THE HORROR.

MMDE wrote:These definitions aren't ambiguous. For example any ring, on the GRE or not, must satisfy the following axioms:
1. It is an abelian group under addition
2. Closed under multiplication
3. Associative under multiplication
4. Obeys the distribution law under multiplication

Now if it has a multiplicative identity, it's known as a ring with identity, if it commutes under multiplication then it's known as a commutative ring. Unless specifically stated (or proven), the above two conditions cannot be assumed.

Similarly, you can find the definitions for any of the other terms you're looking for either online or in a textbook. The beauty of math is, in part, it's rigor in definitions/theorems leaving no room for ambiguity.

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

Posted: Tue Oct 23, 2018 9:34 am
BCLC wrote:I never said ambiguous. I was just want to know what the definition of a ring is in the GRE. Technically the ring definition in the GRE *IS* different from the one in Artin right? This has a few implications. One is that I can't use some of the properties that rings in the Artin book have like how exactly 2 ideals implies field because that property assumes commutativity of rings. Another is the definition of a subring. It's not confusing or ambiguous at all. It's just not explicitly given by ETS.

And that's just for rings! How should I know that what I understand to be 'groups' is what they consider to be 'groups' ? What if ETS' groups are abelian or something? What if ETS' rectangles are actually exclusive rectangles (exclusive of squares) ?

THE HORROR.

That's precisely what I'm saying, the definition for a ring, subring, group...etc are standard almost everywhere (insert measure theory pun here). Some other more obscure topics vary author to author, but for the most part I don't think they'd be that big a discrepancy between texts. To answer your question though, no, you cannot assume all groups are abelian or all rings are commutative on the GRE (or in general for that matter), unless it is stated or able to be proven by the given information .

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

Posted: Wed Oct 24, 2018 12:09 am
MMDE wrote:These definitions aren't ambiguous. For example any ring, on the GRE or not, must satisfy the following axioms:
1. It is an abelian group under addition
2. Closed under multiplication
3. Associative under multiplication
4. Obeys the distribution law under multiplication

Now if it has a multiplicative identity, it's known as a ring with identity, if it commutes under multiplication then it's known as a commutative ring. Unless specifically stated (or proven), the above two conditions cannot be assumed.

Similarly, you can find the definitions for any of the other terms you're looking for either online or in a textbook. The beauty of math is, in part, it's rigor in definitions/theorems leaving no room for ambiguity.

This is so wrong. There is much debate about whether or not the definition of ring should include a unit. As well, while not common, some books don't even assume associativity of multiplication.

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

Posted: Wed Oct 24, 2018 12:29 am
hopeful wrote:
MMDE wrote:These definitions aren't ambiguous. For example any ring, on the GRE or not, must satisfy the following axioms:
1. It is an abelian group under addition
2. Closed under multiplication
3. Associative under multiplication
4. Obeys the distribution law under multiplication

Now if it has a multiplicative identity, it's known as a ring with identity, if it commutes under multiplication then it's known as a commutative ring. Unless specifically stated (or proven), the above two conditions cannot be assumed.

Similarly, you can find the definitions for any of the other terms you're looking for either online or in a textbook. The beauty of math is, in part, it's rigor in definitions/theorems leaving no room for ambiguity.

This is so wrong. There is much debate about whether or not the definition of ring should include a unit. As well, while not common, some books don't even assume associativity of multiplication.

Just because such a debate exists doesn't mean the conventional definition is ambiguous.

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

Posted: Wed Oct 24, 2018 9:31 am
MMDE, thanks for your replies, but why do you keep saying ambiguous? Who said/implied ambiguous? I believe I said/implied non-standard or non-explicit.

MMDE wrote:
hopeful wrote:
MMDE wrote:These definitions aren't ambiguous. For example any ring, on the GRE or not, must satisfy the following axioms:
1. It is an abelian group under addition
2. Closed under multiplication
3. Associative under multiplication
4. Obeys the distribution law under multiplication

Now if it has a multiplicative identity, it's known as a ring with identity, if it commutes under multiplication then it's known as a commutative ring. Unless specifically stated (or proven), the above two conditions cannot be assumed.

Similarly, you can find the definitions for any of the other terms you're looking for either online or in a textbook. The beauty of math is, in part, it's rigor in definitions/theorems leaving no room for ambiguity.

This is so wrong. There is much debate about whether or not the definition of ring should include a unit. As well, while not common, some books don't even assume associativity of multiplication.

Just because such a debate exists doesn't mean the conventional definition is ambiguous.

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

Posted: Wed Oct 24, 2018 9:32 am
You give me hope, hopeful. Thanks!

hopeful wrote:
MMDE wrote:These definitions aren't ambiguous. For example any ring, on the GRE or not, must satisfy the following axioms:
1. It is an abelian group under addition
2. Closed under multiplication
3. Associative under multiplication
4. Obeys the distribution law under multiplication

Now if it has a multiplicative identity, it's known as a ring with identity, if it commutes under multiplication then it's known as a commutative ring. Unless specifically stated (or proven), the above two conditions cannot be assumed.

Similarly, you can find the definitions for any of the other terms you're looking for either online or in a textbook. The beauty of math is, in part, it's rigor in definitions/theorems leaving no room for ambiguity.

This is so wrong. There is much debate about whether or not the definition of ring should include a unit. As well, while not common, some books don't even assume associativity of multiplication.

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

Posted: Wed Oct 24, 2018 12:30 pm
BCLC wrote:MMDE, thanks for your replies, but why do you keep saying ambiguous? Who said/implied ambiguous? I believe I said/implied non-standard or non-explicit.

My apologies, perhaps ambiguous isn't the right word.

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

Posted: Thu Oct 25, 2018 10:37 am
First of all, the question starts off saying "Let $R$ 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 "$R$ 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 $R$ 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 $f$ is increasing". They do not ask questions that could be ambiguous, because that would decrease the reliability of their test.