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Math Help - Commutative and Associative relations of binary operations

  1. #1
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    Commutative and Associative relations of binary operations

    Hi I've got some problems and just need a bit of reassurance that I'm doing them right!

    a) For each of the following binary operations on z^+ say whether or not it is associative, commutative
    i) a*b=a+b^2

    This is neither commutative nor associative

    ii) a*b=a+1

    Is again neither commutative nor associative

    b)Give an example of a binary operation on Z which is commutative but not associative

    I think a*b = a^2+b^2 or a*b=ab+1

    c) On the set of integers Z define a multiplication * by a*b=1+b for any a,b \in Z . Show that <Z,+,*> is not a ring, where + is the usual addition.

    I think this fails because (a*b)*c=1+c, but a*(b*c)= 2+c so not associative under multiplication. However this is for 6 marks so feel like I must be missing something!

    d) Prove the assertion using only axioms of a ring. Let R be a ring with a an element in R. Show there is exactly one element in which a+b=b+a=0.

    For this I assume two distinct elements b and c satisfy this i.e.

    Then (a+b)*c=0, but (b+c)*a=a*b+a*c not zero so assumption fails. Not sure if this is precise enough??

    e) Is it true that every subring in a commutative domain R with unity is an ideal in R? Prove your claim.

    I think this is true but no idea how to prove it

    Sorry I know this is quite a few questions just want a bit of verification that what I'm doing is right and a bit of help with the last question!
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  2. #2
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    You could have added counterexamples to false statements to make it easier for people to check.

    I agree for a) -- c).

    Quote Originally Posted by leshields View Post
    d) Prove the assertion using only axioms of a ring. Let R be a ring with a an element in R. Show there is exactly one element in which a+b=b+a=0.

    For this I assume two distinct elements b and c satisfy this i.e. ???

    Then (a+b)*c=0, but (b+c)*a=a*b+a*c not zero so assumption fails.
    This I don't understand at all. Yes, (a + b) * c = 0, but why a * b + a * c != 0? What is the contradiction?

    Quote Originally Posted by leshields View Post
    e) Is it true that every subring in a commutative domain R with unity is an ideal in R? Prove your claim.
    What about reals as a subring of complex numbers?
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  3. #3
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    Oh yeah that doesn't make sense! and i missed i.e. a+b=0 and  a+c=0

    how about if I say (b+a)+c=b+(a+c) implies  0+c=b+0 hence b=c

    e) Not true!!

    Reals obviously a subring of Complex with unity

    If a\in R and b\in C then R is an ideal of C iff b.a=a.b \in R this is clearly not the case for example taking a=1 and b=i

    Hoping thats a bit better??
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  4. #4
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    Yes, I believe this is correct. I would only replace "iff" in the second last sentence by "only if" because the fact that ba = ab is in R for these particular a and b is not a sufficient condition for R to be an ideal.
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  5. #5
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    Brill, thanks for your help!!
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