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Homework Help: Basic questions about Rings

  1. Jun 27, 2010 #1

    IB1

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    I need to learn some abstract algebra, and it's pretty hard doing this on my own. Please help me.

    According to the definition, Ring is an algebraic structure with two binary operations , commonly called addition (+) and multiplication ( . ). We write (R,+, .). Some examples of rings are: (Z, +, .), (2Z, +, .), (Q, +, .)etc...


    My problem is that where it is written (R,+,.), is + and . normal addition and multiplication respectively, or they're just some binary operations? If they were some binary operations, why they're saying that (Z, +, .) is a ring, without defining + and . ??

    Sorry if my question were silly.
     
  2. jcsd
  3. Jun 27, 2010 #2

    Mark44

    Staff: Mentor

    You can assume that + and . represent the usual operations for addition and multiplication, unless they are defined differently in the problem. (Z, +, .) is the ring of integers, with the usual operations for addition and multiplication.
     
  4. Jun 27, 2010 #3

    IB1

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    Yes, but why then in the begging of the chapter we develop arithmetic on rings if + and . are simply addition and multiplication? Moreover, the third axiom [ c(a+b)=ca +cb, (a+b)c=ac+bc] would be trivial and we wouldn't need to check for it ever because it is always true.

    Sorry for bombing with questions :blush:
     
  5. Jun 27, 2010 #4

    Mark44

    Staff: Mentor

    The distributivity axioms are true for the usual addition and multiplication operations on integers, but for some sets and some operations, they don't hold. The beginning of the chapter is showing you how to verify the axioms on some simple rings.
     
  6. Jun 27, 2010 #5

    IB1

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    Well, I can understand that distributivity might not hold for "some sets and some operations" ... but still it's written + and . and so the operations are addition and multiplication ... I could understand if it were written like this [c#(a*b)=c#a*c#b] ?

    So, Mark44, are you suggesting that when they're writing (R, +, .) they mean that + and . represent some binary operations, and when they write (Z, +, .) they mean normal addition and multiplication ? (and in other cases I've to guess what they mean )

    I still don't get it why they write (R, +, .) instead of (R, #, *). Writing # and * would vanish any doubt or confusion (at least mine). I'm having the same misunderstanding with more than one book.

    Thank you very much for your help
     
  7. Jun 28, 2010 #6

    Mark44

    Staff: Mentor

    Like I already said in post #2, if the addition and multiplication operations are explicitly defined to be different, you can assume that they are the ordinary operations.
     
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