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Abstract algebra: Rings and Ideals

  1. Feb 1, 2010 #1
    1. The problem statement, all variables and given/known data
    The problem is to show that a subset A of a ring S is an ideal where A has certain properties. S is a ring described as a cartisian product of two other rings (i.e., S=(RxZ,+,*)). I have already proved that A is a subring of S and proved one direction of the definition of an ideal. But, the other direction has brought me to having to show that R is commutative. It is given that R is a ring without zero divisors and without identity.


    2. Relevant equations



    3. The attempt at a solution
    I know that a ring R is commutative if it has the property that ab=ca implies b=c when a is not zero. I have attempted various simple manipulations of this statement by using the fact that R is a ring without zero divisors and without an identity.
     
  2. jcsd
  3. Feb 1, 2010 #2
    In other words, is a ring R without zero divisors and without an identity commutative.
     
  4. Feb 1, 2010 #3
  5. Feb 1, 2010 #4
    An integral domain has a unity (i.e., identity). In my case, R has no unity so it is not an integral domain.
     
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