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

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    I have to following axioms:

    A1)  a+b  \epsilon  R,  \forall a,b  \epsilon  R

    A2)  a+b = b+a, \forall a,b \epsilon R

    A3)  (a+b)+c = a+(b+c), \forall a,b,c \epsilon R

    A4)  \exists 0 \epsilon R s.t, a+0 = a = 0+a, \forall a \epsilon R

    A5)  \exists b \epsilon R s.t, a+b = 0 = b+a, \forall a \epsilon R


    M1)  a \cdot b \epsilon R, \forall a,b \epsilon R

    M2)  a \cdot b = b \cdot a, \forall a,b \epsilon R

    M3)  (a \cdot b) \cdot c = a \cdot (b \cdot c), \forall a,b,c \epsilon R

    M4)  \exists 1_{R} \epsilon R s.t, a \cdot 1_{R} = a = 1_{R} \cdot a, \forall a \epsilon R

    M5)  \exists b \epsilon R s.t, a \cdot b = 1_{R} = b \cdot a, \forall a \epsilon R


    D1)  a \cdot (b+c) = a \cdot b + a \cdot c, \forall a,b,c \epsilon R

    D2)  (a+b) \cdot c = a \cdot c + b \cdot c, \forall a,b,c \epsilon R

    Z)  If, a \cdot b = 0, then, a = 0, b = 0, or, a = b = 0, \forall a,b \epsilon R


    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!)
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    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 a \not= 0, what is a^{-1}ab?

    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.
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    (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 a \not= 0, what is a^{-1}ab?

    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 a \not= 0, what is a^{-1}ab? 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
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    (Original post by adie_raz)
    if ab=0 and a \not= 0, what is a^{-1}ab? 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
    :yep: I think the article is right. a^{-1} a b = b so b=0 and Z can be deduced to be true.
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    (Original post by sonofdot)
    :yep: I think the article is right. a^{-1} a b = b so b=0 and Z can be deduced to be true.
    Thanks
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    (Original post by adie_raz)
    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.
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    (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
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    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...
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    (Original post by adie_raz)
    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}
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    So for a field we have every NON-ZERO element is invertible, not every element is invertible, therefore fields contain zero as an element??
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    (Original post by adie_raz)
    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.
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    (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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