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# What are the axioms of a ring and field?? watch

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1. I have to following axioms:

A1)

A2)

A3)

A4)

A5)

M1)

M2)

M3)

M4)

M5)

D1)

D2)

Z)

I want to know which ones you need to CHECK to make sure something is a field of ring.

Wiki says you need A1) - A5), M1), M3), M4), D1) and D2) for a ring and for a field you need the previous axioms along with M2) and M5).

My problem is my lecture notes do not mention closure (which I think is incorrect) also neither does my textbook (Rings, Fields and Groups: An Introduction to Abstract Algebra - I think this again is wrong, but makes me doubt myself more) also in the textbook, it says that you need Z for a field. Is this true or does this just become true due to the presence of the other axioms??

Please remember I want the axioms, and therefore the things you must check to see if a set is a Ring/Field. If for example Z is true, but is just induced due to the presence of the other axioms, please state that.

Thanks for any help, rep will be given!

(P.S. Please excuse my LaTeX, I didn't put any spaces in!)
2. Z is the condition for a ring to be an integral domain. It can be deduced from the existence of a multiplicative inverse in fields (M5) - if ab=0 and , what is ?

Obviously you do need closure for a ring, but I think having well-defined operations implies closure, so you don't always need to write it down.
3. (Original post by sonofdot)
Z is the condition for a ring to be an integral domain. It can be deduced from the existence of a multiplicative inverse in fields (M5) - if ab=0 and , what is ?

Obviously you do need closure for a ring, but I think having well-defined operations implies closure, so you don't always need to write it down.
if ab=0 and , what is ? 0

Thank you for your answer, but just to be clear are you saying that wiki is correct??

The link for a ring is:

http://en.wikipedia.org/wiki/Ring_(mathematics)

and for a field is:

http://en.wikipedia.org/wiki/Field_(mathematics)

Thanks
if ab=0 and , what is ? 0

Thank you for your answer, but just to be clear are you saying that wiki is correct??

The link for a ring is:

http://en.wikipedia.org/wiki/Ring_(mathematics)

and for a field is:

http://en.wikipedia.org/wiki/Field_(mathematics)

Thanks
I think the article is right. so b=0 and Z can be deduced to be true.
5. (Original post by sonofdot)
I think the article is right. so b=0 and Z can be deduced to be true.
Thanks
My problem is my lecture notes do not mention closure (which I think is incorrect) also neither does my textbook (Rings, Fields and Groups: An Introduction to Abstract Algebra - I think this again is wrong, but makes me doubt myself more) also in the textbook, it says that you need Z for a field. Is this true or does this just become true due to the presence of the other axioms?
For closure, you probably have the operations defined as functions RxR -> R. That ensures closure so it's not given as a numbered axiom. But you still need to check it. Eg, if you're showing that some set of matrices is a ring, you need to multiply two of them together to check the result still has the same form and hence show that multiplication really is a function RxR -> R.
7. (Original post by SsEe)
For closure, you probably have the operations defined as functions RxR -> R. That ensures closure so it's not given as a numbered axiom. But you still need to check it. Eg, if you're showing that some set of matrices is a ring, you need to multiply two of them together to check the result still has the same form and hence show that multiplication really is a function RxR -> R.
Thanks very much, indeed the operations are defined as RxR -> R
8. Another question on the same lines:

F* = F \ {0} correct??

This is supposed to be true, but due to axiom M5), surely this F* = F as all elements of F are already invertible...
Another question on the same lines:

F* = F \ {0} correct??

This is supposed to be true, but due to axiom M5), surely this F* = F as all elements of F are already invertible...
For a ring, R* is often used to denote the set of invertible elements of R; for a field F = R, since every non-zero element is invertible, we have that F* = F\{0}
10. So for a field we have every NON-ZERO element is invertible, not every element is invertible, therefore fields contain zero as an element??
So for a field we have every NON-ZERO element is invertible, not every element is invertible, therefore fields contain zero as an element??
Your conclusion is correct, but your reasoning is ... roundabout. Fields have a zero element simply because fields are a special kind of ring.
12. (Original post by Zhen Lin)
Your conclusion is correct, but your reasoning is ... roundabout. Fields have a zero element simply because fields are a special kind of ring.
Thank you for confirming

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