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

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

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?!

---

I asked on maths SE:

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

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?!

---

I asked on maths SE:

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?

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

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.

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?

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?

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.

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 aring with identity, if it commutes under multiplication then it's known as acommutative 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?

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 .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.

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

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.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 aring with identity, if it commutes under multiplication then it's known as acommutative 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?

Just because such a debate exists doesn't mean the conventional definition is ambiguous.hopeful wrote: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.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 aring with identity, if it commutes under multiplication then it's known as acommutative 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?

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

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 aring with identity, if it commutes under multiplication then it's known as acommutative 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?

You give me hope, hopeful. Thanks!

hopeful wrote:

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 aring with identity, if it commutes under multiplication then it's known as acommutative 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?

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

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

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 $$R$$ 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

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

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.

Hope this helps.

https://www.mathsub.com/gr0568-66/

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.

Hope this helps.